IN  MEMORIAM 
FLOR1AN  CAJORI 


<U# 


— •   ,  * 


IMPROVED  ABACUS: 


AN    EXPLANATORY    TREATISE 


THEORY   AND    PRACTICE 


ARITHMETIC    AND   MENSURATION, 


BY   THOMAS    RAINEY. 

ENERGY     IS    THE    PRICE    OF     SUCCESS. 

CINCINNATI: 

PUBLISHED    BY    E.  D.  TRUMAN 

111,  MAIN  STREET. 

1849. 


RECOMMENDATIONS. 

From  the  Cincinnati  Evening  Nonpareil. 

"  We  have  received  a  copy  of  '  Rainey's  Improved  Abacus,'  a  treatise  on  Arithmetical 
calculations,  and  take  pleasure  in  recommending  it  as  a  work  deserving  the  patronage  of 
business  men  desiring  a  knowledge  of  a  plain,  concise,  and  practical  system.  It  simpli- 
fies and  shortens  calculations  bv  canceling  or  expunging  the  numbers,  and  is  of  incalcula- 
ble benefit  to  those  whose  business  requires  of  them  a  simplified  method  of  calculation." 

From  the  Cincinnati  Daily  Evening  Times. 

'•  Messrs.  E.  I).  Truman  &  Co.  have  just  issued  a  Treatise  on  Arithmetic  by  the  can- 
celing method.  The  work  is  by  Prof.  Rainey,  and  is  well  prepared  for  popular  instruction. 
All  questions  in  numbers  are  of  either  Multiplication  or  Division,  and  this  is  the  base  of 
the  system.  We  can  truly  say,  the  canceling  method  is  a  superior  one  for  all  ordinary 
calculations,  and  we  trust  it  will  be  generally  cultivated.  Mr.  Rainey  has  given  to  the 
public  his  analysis  in  cheap  form,  and  they  are  under  great  obligations  therefor." 

From  the  Cincinnati  Great  West. 

"THE  IMPROVED  ABACUS:  an  explanatory  treatise  on  the  Theory  and  Practice  of 
Arithmetic  and  Mensuration,  in  which  the  general  principles  involved  in  practical  calcu- 
lations are  thoroughly  elucidated  and  illustrated  by  numerous  analytic  and  abbreviated 
examples.  By  Thomas  Rainey.  Energy  is  the  price  of  success." 

"  This  is  the  title  of  a  neat  volume  from  the  press  of  Messrs.  Truman  &  Co.  The  design 
of  the  work  is  to  give  to  practical  men  and  advanced  students  a  thorough  theoretical  and 
practical  knowledge  of  all  such  calculations  as  arc  necessary  in  commercial  and  mechani- 
cal transactions;  and  this,  too,  on  common  sense  principles  ;  '  for,'  says  the  author, '  rules 
without  reasons  are  ridiculous,  and  insulting'  to  the  inquiring  mind.'  That  a  work  ot 
this  character  has  long  been  needed,  no  practical  man  can  deny  ;  and  every  advanced 
student  in  the  science  will  hail  its  appearance  with  delight. 

"  Knowing  practically,  the  many  deficiencies  of  the  old  systems  used,  we  cannot  too 
highly  recommend  this  excellent  treatise." 

From  Prof.  WM.  McGoomN,  Sidney,  O. 

"  I  have  examined  a  small  work  published  by  Prof.  T.  Rainey,  on  Interest,  Profit  and 
Loss,  Proportions,  Discounts,  Insurance,  Stocks,  Commission,  Mensuration,  etc.,  and  am 
much  pleased  with  it.  It  is  just  such  a  work  as  the  learner  needs  ;  it  supersedes  the  ne- 
cessity of  a  teacher.  The  roles  are  concise  and  practical. 

WiM.    MC'GOOKIN." 

From  Prof.  PRATT,  University  of  Missouri. 

"  Having  attended  your  course  of  lectures,  in  Arithmetic,  etc.,  you  will  allow  me  to  ex- 
press to  yon  my  entire  conviction  that  your  system  is  peculiarly  adapted  to  the  simplifica- 
tion of  Arithmetical  calculations  generally,  and  to  the  abbreviation  of  most  intricate 
computations.  Very  respectfully,  &c.,  GEO.  C.  PRATT." 

"  I  fully  concur  with  Prof.  PRATT.  WM.  T.  DAVIS, 

Columbia  High  School." 

From  H.  F.  WEST,  of  Indianapolis. 

*****"  The  unquestionable  accuracy,  as  well  as  the  facility  of  Prof. 
Rainey's  process  of  solving  problems,  commends  itself  to  the  plain  common  sense  of 

every  "observer,  as  it  is  so  happily  adapted  to  the  practical  purposes  of  life The 

ready  manner  by  which  the  results  of  intricate  questions  were  obtained,  correctly  too, 
showed  that  Prof.  Rainey  had  happily  adapted  the  principles  of  his  improvements  to  all 
those  perplexing  rules  that  fill  up  our  school  Arithmetics.  *  *  *  Every 

business  man,  and  every  young  man  that  intends  being  one,  should  not  be  unacquainted 
with  this  important  attainment  in  the  science  of  numbers." 

From  Prof.  McWiLLiAMS,   Springfeld,  O.,  late  Presiding  Officer  of  Ohio 

School  Convention. 

*  *  *  *  "  He  lectures  at  present  to  our  school  at  large,  and  from  this  test,  I  find 
him  to  be  what  he  professes.  *  *  Do  not  fail,  if  practicable  at  all,  to  try  his 

system  ;  and  the  farther  the  better." 


From  the  Cincinnati  Gazette. 

.  .  .  .  "  The  author  commences  by  showing  that  if  a  quantity  is  to  be  multiplied  and  divi- 
ded by  the  same  number,  it  will  remain  unchanged  in  value  ;  and  consequently  neither  op- 
eration need  be  performed  :  Or,  if  a  quantity  is  to  be  multiplied  by  a  given  number 
and  divided  by  another  number  half  as  great,  we  may  in  the  first  instance,  multiply  by  half 
the  number,  and  neglect  the  division  altogether,  and  the  result  will  be  the  same  that  it  would 
have  been,  had  we  performed  both  operations,  and  gone  through  with  twice  the  amount 
of  labor.  The  author  does  not  confine  himself  to  the  simple  examples  we  have  given, 
but  shows  that  this  great  principle  —  that  PlHJOsing  forces  _ojLftqu.al  strength  will  destroy 
each  otlie_tr-runs  through  the  whole  systeMf,  and^  thal^it  may  be  used  to"  immense  advan- 
tage under  all  of  its  rules  ;  in  fact,  it  is  the  main  pillar  upon  which  the  author  has  reared 
his  beautiful  system  .....  What  next  strikes  the  reader  of  this  instructive  work,  is, 
that  the  author  permits  no  opportunity  of  imparting  instruction  to  pass  by  him  unimproved. 
The  terms  of  art  which  he  uses  are  all  clearly  and  beautifully  explained  ;  and  the  mechan- 
ical and  tedious  methods  of  instruction  adopted  by  many  of  the  old  writers,  are  ex- 
posed." 

From  the  Tennessee  [Nashville]  Organ  —  Rev.  JOHN  P.  CAMPBELL,  Editor. 

NEW  AND  USEFUL  BOOK'  FOR  EVERYBODY. 


RAINEY'S  IMPROVED  ABACUS  :    AN  EXPLANATORY  TREATISE  ON  THE  THEORY 
AND  PRACTICE  OF  ARITHMETIC  AND  MENSURATION. 

THE  work  above  noticed,  was  written  with  direct  reference  to  the  wants  of  all  classes 
of  business  men.  The  system  is  new,  very  short,  and  beautiful. 

Experience  teaches  the  business  man  that  the  old  system  requires  too  many  figures  :  that 
the  rules  are  too  tedious  and  arbitrary  :  that  in  too  .many  instances  results  are  found  by  the 
arbitrary  and  incomprehensible  arrangements  of  the  books  ;  and  that  if  he  is  unable  to  re- 
tain or  apply  such  rules,  he  must  fall  back  on  common  sense,  and  construct  his  own  rules. 
Now,  all  rules,  without  reasons,  we  consider,  ridiculous  and  insulting  to  the  intelligent  and 
inquiring  mind  :  but  with  reasons  presented  first,  and  afterward  calculations  to  illustrate 
their  application  and  use,  it  becomes  an  easy  matter  for  any  person  of  common  mind  to 
make  his  own  rules.  Hence,  every  rule  in  Arithmetic  and  Mensuration  is  fully,  clearly, 
and  satisfactorily  explained  ;  while  a  great  number  of  examples,  such  as  occur  in  every- 
day business,  are  wrought  out;  thus  enabling  the  practical  man  to  find  in  the  book  the 
method  of  making  calculations  similar  to  any  that  he  may  find  necessary. 

The  practical  calculator  will  find  in  this  work  a  system  of  calculations  extraordinarily 
short,  simple,  and  satisfactory;  and  explained  in  PLAIN,  FAMILIAR  LANGUAGE, 
free  from  difficult  terms,  which  too  often  obscure  the  sense,  without  satisfying  the 
judgment. 

No  department  of  calculations  is  omitted  in  this  work,  which  would  be  of  interest  to 
business  men  and  students,  of  any  trade  or  profession. 

MERCHANTS  and  COMMERCIAL  men  will  find  everything  relating  to  their  voca- 
tions, and  a  great  variety  of  new  and  interesting  matter,  not  before  introduced  into  any 
work  of  this  kind.  We  invite  particular  attention  to  the  system  of  Interest,  Discount, 
Profits  and  Losses,  the  several  varieties  of  Commission,  Insurance,  etc.  ;  the  combina- 
tion of  several  different  statements  in  one  statement  ;  and  to  the  Tables  for  Banking, 
Equation,  etc. 

MECHANICS  are  invited  to  examine  the  great  variety  of  work  pertaining  to  MA- 
CHINERY ;  the  MECHANICAL  POWERS  ;  the  TABLES  of  AREAS,  etc.  ;  WEIGHTS  of 
METALS,  etc.  ;  Circles,  Cylinders,  Globes,  Balls,  etc.  ;  Contents  and  Weights  of  Solid 
.Bodies,  and  all  of  the  Superficial  and  Solid  Measurements  necessary  in  any  department 
of  mechanics. 

The  attention  of  SCIENTIFIC  men,  is  directed  to  the  new  and  beautiful  system  of 
PROPORTIONS,  CAUSE  and  EFFECT  ;  the  Philosophy  of  the  general  Method  of  Statement 
throughout  the  whole  work  :  COMBINATION  of  STATEMENTS  ;  COMPLEX  FRACTIONS  ;  the 
general  and  easy  method  of  disposing  of  Fractions,  etc.,  etc. 

It  has  been  the  constant  aim  of  the  author  to  combine  UTILITY,  BREVITY,  and  SIMPLI- 
CITY ;  to  use  such  language  in  all  explanations,  as  could  be  easily  understood  ;  and  to 
present  to  the  public  a  real  improvement. 

All  such  persons  as  wish  to  learn  a  short,  certain,  and  easy  method  of  making  general 
business  calculations,  without  the  assistance  of  a  Teacher,  will  find  in  this  work  every- 
thing that  is  calculated  to  assist  them  in  this  desirable  undertaking.  ASSISTANCE  TO  THE 
PRIVATE  STUDENT  is  a  peculiar  feature  of  this  treatise. 

Several  hundred  COIN  PLATES  arc  added,  of  whose  utility  it  'is  unnecessary  to  say 
anything  to  those  concerned. 

"It  is,  without  doubt,  a  very  useful  book."    Cincinnati  Daily  Commercial. 


RECOMMENDATIONS. 

From  the  Wiyne  County  Whig. 

"  We  again  call  the  attention  of  our  readers  to  the  appointment  of  Prof.  Rainey,  to 
lecture  at  the  Methodist  Church,  on  this  Wednesday  evening.  We  had  the  pleasure  of 
attending:  the  lectures  delivered  in  our  place  by  this  gentleman  last  week.  Under  ordinary 
dircnmBtanees,  the  subject  of  Mathematics  is  exceedingly  dry  and  uninteresting.  To  the 
lectures  of  Mr.  Rainey,  we  listened  with  most  intense  interest,  as  did  all  who  heard  him. 
jives  to  the  sul.jrct  a  novelty  and  interest  we  have  never  before  witnessed.  His  new 
mode  of  calculation  is  the  shortest,  most  rapid,  and  simple  of  any  mode  now  in  use.  We 


entertain  the  very  liiirlu^t  opinion  of  the  practical  utility  of  his  plan,  and  commend  it  to 
the  favorable  notice  of  students  and  business  men.  The  following  communication  was 
sent  us  for  publication.  We  indorse  readily  and  fully  the  commendations  it  contains.  It 
comes  from  men  well  known  as  worthy  of  every  confidence,  and  well  qualified  to  judge  of 
what  they  speak. 

"  MR,  EDITOR  :  —  Prof.  Rainey,  who  has  been  lecturing  in  our  place  some  days,  on  his 
short  and  beautiful  system  of  practical  calculations,  intends  visiting  your  town.  We 
take  pleasure  in  commending  this  matter  to  your  attention,  because  we  feel  convinced 
that  he  will  greatly  benefit  all  who  go  to  hear  him.  In  his  lectures  in  this  place,  he  has 
proved  his  system  of  calculations  to  be  the  shortest  and  the  most  simple,  as  well  as  the 
most  expeditious  that  we  have  ever  seen.  His  operations  at  the  blackboard  convince 
every  thinking  man  of  brevity  and  precision,  certainty,  simplicity,  and  final  satisfaction  ; 
while  they  excite  equal  wonder  and  pleasure  for  their  philosophical  beauty  and  grandeur. 
He  throws  away,  instead  of  using  figures,  and  works  all  fractional  questions  with  ease  and 
simplicity. 

"  Nothing  that  we  have  ever  seen  can  equal  his  calculations  in  Simple  Interest.  Instead 
of  a  dozen  different  rules  for  different  rates  per  cent.,  he  has  one  simple  rule,  short  and 
easily  understood,  for  working  questions  of  any  conceivable  principal,  time,  and  rate  per 
cent.  This  every  practical  man  knows  to  be  a  great  convenience.  Prof.  Rainey  has 
with  him  a  large  work  in  which  his  system  is  thoroughly  developed  ;  treating  of  every  va- 
riety of  calculation  that  can  possibly  occur  among  business  men,  either  in  Commercial  and 
Mechanical  Arithmetic,  or  in  Mensuration.  This  is  an  excellent  work,  and  the  first  we 
have  seen  that  thoroughly  explained  these  operations.  He  (jives  his  reasons  for  a  thing  first; 
then  deduces  the  rule.  Hence,  his  rules  appeal  to  Common  sense.  His  work  in  these  rules, 
as  well  as  others,  cannot  fail  to  excite  admiration  and  attention  wherever  intelligent  men 
will  take  time  to  investigate.  As  he  has  but  a  short  time  to  stay  in  your  place,  we  think 
it  well  to  call  the  attention  of  your  citizens  to  the  subject,  that  they  may  secure  the  op- 
portunity, and  not  regret  when  too  late.  Let  all  your  people  hear  him,  and  not  fail  to  se- 
cure his  books. 

[Signed  ]        JAMES  M.  POE,      \ 

CHARLES  FISKE,     >  Teachers,  Richmond,  la. 
WILLIAM  AUSTIN,  ; 

DR.  JOHN  PRICHET,)  n     ,       .,,      T    „ 
REV.  E.  MCCHORD,!  Centreville,  la." 

*  *  *  *  "  RESOLVED,  11.  That  Prof.  Rainey's  book  is  just  such  a  work  as 
the  learner  needs  ;  because  every  principle  and  operation  is  so  thoroughly  explained  and 
illustrated,  that  by  its  investigation,  the  ordinary  reader  will  be  enabled  to  comprehend 
and  practice  the  system. 

"  RESOLVED,  12.  That  his  system  of  Interest,  as  taught  in  his  work,  is  of  itself,  worth 
more  than  his  charge  for  both  instruction  and  his  books,  and  being  short  and  easily  under- 
stood, is  pre-eminently  adapted  to  the  counting-house. 

[Signed  by]  P.  Y.  Wilson,  Finley  Bigger,  John  W.  Barber,  Dr.  W.  Frame,  J.  A. 
Kendall,  Dr.  A.  Norris,  and  others  of  Rushville,  la  ;  by  John  C.  Osborn,  Thomas  Kirby, 
G.  B.  Holland,  Dr,  Andrews,  Esquire  Swarr,  George  B.  Norris,  and  others,  of  Muncie,  la  ; 
and  by  Win.  F.  Kelso,  James  McMeans,  and  twenty  others,  of  Newcastle,  la." 

From  the  Cincinnati  Evening  Dispatch', 

also, 

From  the  Cincinnati  Daily  Enquirer. 

"  RAINEY'S  IMPROVED  ABACUS.  ...  It  purports  to  be  «  An  explanatory  Treatise 
on  the  Theory  and  Practice  of  Arithmetic  and  Mensuration  ;'  and  from  a  hasty  examina- 
tion it  appears  to  be  of  great  practical  use  in  making  calculations  in  a  great  variety  of 
business  relations.  It  is  recommended  in  high  terms  by  teachers  and  others  competent  to 
judge,  who  have  examined  it." 


HAINEY'S   IMPROVED   ABACUS; 


AN  EXPLANATORY  TREATISE  ON  THE 


THEORY    AND    PEACTICE 


OF 


AEITHMETIC   AND   MENSURATION: 


IN   WHICH    THE    GENERAL    PRINCIPLES    INVOLVED   IN    PRAC- 
TICAL   CALCULATIONS   ARE    THOROUGHLY    ELUCI- 
DATED AND  ILLUSTRATED  BY  NUMEROUS 
ANALYTIC    AND   ABBREVIATED 
EXAMPLES. 


BY     THOMAS    R  A  I  1ST  E  Y . 


ENERGY   IS  TH!i  P1UC1?  OF  SUCCESS. 


CINCINNATI: 
E.    D.   TRUMAN,  PUBLISHER. 

1849 


NOTE  TO  TEACHEKS. 

The  Teacher  will  observe  that  this  work  is  devoted  exclusively  to  the  de- 
velopment and  illustration  of  the  principles  of  numbers,  with  the  introduc- 
tion of  such  a  number,  and  variety  of  examples,  only,  as  subserve  this  pur- 
pose. It  is  deemed  the  privilege  of  the  teacher,  to  present  such  examples 
for  practice  and  test,  as  may  best  accord  with  his  judgment,  and  in  the 
highest  degree  develop  the  capacities  of  the  learner. 

This  arrangement  presupposes  the  plan  of  instructing  classes  at  the 
blackboard,  by  familiar  illustrations,  and  the  occasional  test  of  each  pupil's 
progress,  in  the  presence  of  the  whole  class.  A  blackboard  isindispensa~ 
ble  to  every  good  school. 

In  Mensuration,  every  figure  explained  in  the  text  should  be  carefully 
drawn  on  the  blackboard;  that  the  twofold  purpose  of  illustration  and 
draining  might  be  subserved  at  the  same  time. 

It  has  been  deemed  useless  to  treat  of  the  Elements  of  Arithmetic,  af 
this  department  of  numbers  is  generally  studied  in  a  separate  book;  and 
as  a  considerable  number  of  good  elementary  works  now  claim  the  patron- 
age of  the  public. 


Entered  according  to  Act  of  Congress,  in  the  year  1849, 

By  THOMAS  RAINEY, 
In  the  Clerk's  Office  of  the  District  Court,  for  the  District  of  Ohio. 


C.  MORGAN  &  Co.,  Stereotypers. 
MORGAN  &  OVEREND,  Printers. 


PREFACE. 


IT  would  be  both  absurd  and  arrogant  for  an  author,  at  this 
day,  to  present  a  new  text-book  on  Arithmetic,  according  to 
the  old  and  long  cherished  standards.  A  large  number  of  ac- 
complished mathematicians  and  experienced  teachers,  as  well 
in  our  own  country  as  in  Europe,  have  plied  their  talents  and 
energies  to  this  subject  with  peculiar  ability  and  success;  pre- 
senting works,  perfect  in  their  order  and  arrangements,  and 
sufficiently  intelligible  and  satisfactory,  so  far  as  the  old  sys- 
tems are  concerned,  for  all  practical  purposes. 

_     Although  Arithmetic  has,  until  recently,  been  neglected  by 
scienlific^ttfiii,  yet  the  onward  progress  of  latter-day^improve- 

"ments  has  necessitated  a  corresponding  advance  in  this  sci- 
ence; until  a  place  is  now  conceded  it  among  the  rational  and 
explicable  sciences,  instead  of  among  the  mere  handicraft  arts, 
as  hitherto.     This  advance  in  the  principles  of  the  science, 
has  involved  new  issues.     It  is  found  that  the  ordinary  system     . 
of  statements  is  too  mechanical)  circumstantial,  and  uncertain]     • 
that  the  method  of  statement,  within  itself,  precludes  that  ra- 
tional analysis,  and  thorough  demonstration  of  principles  neces- 
sary to  the  proper  appreciation  and   use  of  any  science;  and 
that  a  sum  of  labor  is  performed,  intne  practical  reduction    " 
of  calculations,  which  is  not  only  unnecessary,  but  which  di-   ;    - 
verts  the  mind  from  the  proper  issues  involved  in  the  pro- 
blem, and  leads  it  into  a  labyrinth  of  doubts  and  obscurities. 
It  is  to  the  remedy  of  these  palpable  defects,  that  our  labors 
are  addressed;  and  to  the  presentation  of  a  rational,  satisfac- 
tory, simple,  unique  and  brief  system  of  calculations,  such  as 
demanded  by  the  practice  of  ordinary  business  transactions. 

It  is  likewise  designed  to  furnish  the  private  student  with  an 
easy  and  certain  guide  to  proficiency  in  numbers,  on  such 

(iii) 


IV  PREFACE. 

principles,  and  in  sucli  accordance  with  common  sense,  as  will 
appear  to  him  reasonable  and  convincing.  Hence,  while  the 
theory  of  each  department  of  numbers  treated,  is  clearly  elu- 
cidated, the  practice  is  illustrated  and  demonstrated  by  familiar 
examples,  occurring  in  business;  and,  too,  by  a  method  so 
short  and  general  in  its  application,  as  to  admit  of  demonstra- 
tion more  conclusive  and  pointed,  than  when  encumbered  by 
the  masses  of  unnecessary  multiplication,  division,  etc.,  which 
always  attend  the  solution  by  the  ordinary  method. 

Throughout  the  whole  treatise,  the  reader  will  observe  that 
much  importance  is  attached  to  the  comprehension  and  proper 
application  of  the  principles  of  Proportion;  as  constituting, 
chiefly,  the  basis  of  all  those  operations  which  follow  the  ele- 
mentary rules.  In  accordance  with  this  view,  all  of  the 
statements  in  this  work,  except  those  of  addition  and  sub- 
traction, are  made  by  Simple,  Compound,  or  Concatenated 
Proportion;  hence,  they  are  unique,  and  may  be  easily 
remembered.  - 

The  beautiful  theory  of  Cause  and  Effect  is  thoroughly 
discussed,  and  applied  to  Compound  Proportion.  The  theory 
of  Inverse  Proportion  will,  as  dependent  on  Cause  and  Effect, 
be  found  quite  different  from  any  hitherto  presented. 

Much  attention  has  been  given  to  Mensuration,  because  of 
its  practical  utility,  and  the  constant  necessity  of  the  appli- 
cation of  its  principles  in  active  life. 

Contrary  to  the  custom  of  many  authors,  we  have  excluded 
from  this  treatise  all  of  those  sub-divisions  of  numbers  whose 
explanation  depends  on  algebraic  principles;  such  as  the  Po- 
sitions, Alligation,  the  Progressions,  Permutation,  Cube  Rootj 
etc.;  none  of  which  offer  any  reward  for  the  arduous  labor 
lost  in  the  impossible  task  of  their  attainment  in  Arithmetic. 
Jf  one-half  the  time  devoted  to  these  principles,  in  their  arbitrary 
form  in  arithmetic,  were  given  to  the  study  of  Algebra,  the  pupil 
would  not  only  learn  a  great  portion  of  that  beautiful  science, 
but  would  thus  secure  the  only  key  to  the  principles  involved  in 
these  rules. 

It  has  been  a  prime  object,  first,  to  discuss  principles,  and 


PREFACE. 


then  deduce  practical  directions;  for  rules,  wUJiout  reasons,  are  ^\»  /v  /, 
ridiculous,  and  insulting  to  the  inquiring  mind.  I  W  */ 

No  secondary  principle  has  been  used  in  the  elucidation  or 
illustration  of  one  that  is  primary;  nor  has  any  princi- 
ple been  anticipated;  but  each,  used  in  its  natural  sequence, 
has  been  made  the  basis  of  a  yet  higher  principle,  in  such 
manner,  as  to  cultivate  the  reasoning  powers  of  the  learner,  with-  \ 
out  embarrassing  them. 

J£echnical  phraseology  has  been  avoided,  as  far  as  consistent 
with  the  requirements  of  such  a  treatise;  a-s  likewise,  puzzles, 
and  all  giddy  theorizing  on  trivial  and  unimportant  topics, 
which  should  be  beneath  the  dignity  of  a  scientific  man,  al- 
though well  calculated  to  please  the  fancies  of  a  vacant  mind: 
for  he  who  would  be  a  useful  man,  must  be  a  practical  man; 
and  the  less  acquainted  with  fascinating  chimeras,  the  better  he 
is  adapted  to  his  great  purpose. 

It  is  not  claimed  for  this  system  that  cancelation  can  be 
availed  in  every  solution;  but  that  a  great  majority  of  prac- 
tical questions  can  be  much  abbreviated  by  it;  the  excellency 
claimed  for  the  system,  is,  that  while  it  abbreviates  the  work, 
the  statement  is  so  simple,  so  philosophical,  and  the  result,  so  I 
inevitable,  that  no  intelligent  individual  can  fail  practicing  its  t 
principles,  whether  the  arbitrary  rules  be  remembered  or  not 

We  shall  endeavor  to  deal  mildly  with  those  who,  being 
bound  to  the  old  system,  as  their  hobby,  cannot,  or  will  not,    \  . 
epen  their  eyes  to  the  evident  advance  of  modern  improve-    Uu  / 
ments;  and,  therefore,  submit  our  labors  to  the  investigation   * 
of  those  candid  and  intelligent  minds,  which  are  not  shackled  //    / 
down  to  such  usages  of  the  past,  as  are  endeared  more  by^    /, 
habit,  than  by  any  rational  merit  /     / 

T.  RAINEY. 

^Cincinnati,  July,  1849. 


INDEX. 


PAGE. 

Theory  of  Cancelation, 5 

Rule  for  Cancelation, 7 

Common  Fractions, 8 

Multiplication  of  Fractions,    9 

Division  of  Fractions, 10 

Rule  for  Fractions, 11 

Complex  Fractions, 11 

Reduction  of, 12 

Multiplication  of, 14 

Division  of, 15 

Ratio  and  Proportion, 47 

Definition  of  Proportion,  and 
new  method  of  Statement 

and  Solution, 48 

Analysis  in  Proportion,  ....  50 

Contrast  of  old  and  new  systems,  51 
Obscurity    of    the    ordinary 

form  of  Statement, 55 

Statements  and  Solutions,..  57 
Ex.  in  Complex  Fractions,. .  61 
Rule  for  Direct  Proportion,. .  62 

Simple  Interest, 16 

4         Theory  of, 17 

Rule  for  all  Rates, 25 

Interest  at  6  per  cent., 28 

Illustration  of  Theory ,. ....  32 
Rule  for  6  per  cent., 35 

Interest  at  7  per  cent., 36 

Rule  for  7  per  cent., 39 

Interest  at  8  per  cent., 37 

Interest  at  12  per  cent., 38 

Rule  for  8  and  12  per  cent.,. .  39 

Rate  and  Forfeiture  Table  in  Int.  39 

Law  of  making  and  transferring 

Notes,.! 37 

Lapse  of  time  between  two  Notes,  39 

Partial  Payments, 40 

Ohio,     Indiana,    and     Ken- 
tucky Rule, 40 

Calculation  illustrating  Rule,...  42 
U.  S.  Supreme  Court  rule, . .  45 
Commercial  or  Vermont,  and 
Connecticut  rule, 46 

Discount:  Theory, 95 

Rule  for  Discount, 99 

True  and  False  Discount,..  100 

To  find  the  Face  of  a    Note,  to 
diaw  a  Specific  Sum, ......  101 

(vi) 


PAGE. 

Equation  of  Payments, ......  t ..  171 

Rule  for  Equation, 173 

Time    Table    for     Banking   and 

Equation, 278 

Profit  and  Loss, 62 

Discussion  of  General  The- 
ory,    63 

Five  Varieties  of, 64 

Var.  1— To   find   the   selling 
price  at  a  given  per  cent. 

gain  or  loss, 66 

Var.  2— To  find  the  rate  per 
centum   profit   or   loss    on 

sales, 72 

Var.3 — To  find  the  Cost  Price 
after    gaining  or  losing  a 

given  per  cent., 75 

Var.  4 — Compound  Profit  and 

Loss, 81 

Var.  5— Combination  of  state- 
ments,    85 

Examples  in  Combination, . .  88 
Miscellaneous  examples,...  92 
Rule  for  Combining  several  Oper- 
ations in  one  Statement,  .  94 

Commission, 102 

Two  methods  of  deducting, .  105 

Brokerage  and  Stocks, 105 

Statements  combined, 107 

Rule  for  combined    Broker- 
age, Stocks,  etc.,... 107 

Insurance, 108 

Four  Varieties, 109 

Discussion  of  propriety  of  In- 
surance,  Ill 

Var.  1,  and  Rule, 113 

Combination  of  Statements,  115 

Var.  2,  and  Rule, 116 

Var.3,  and  Rule, 117 

Var.  4,  and  Rule, 118 

Life  Insurance, 119 

Tolls  and  Rule, 120 

Compound  Proportion, 123 

Theory  of  Cause  and  Effect: 
all  animate  things  Causes,126 

Causes  of  Time, 126 

Geometrical  Extent  a  Cause,127 
Capital  aCause» 127 


INDEX. 


vii 


PAGE. 

Classification  of  Causes  in  the 

statement, 129 

The  Causes  and  Effects,the  four 
terms  of  a  geometrical  pro- 
portion,   131 

Ratio  among  Causes, 132 

Examples  and  Proofs, 134 

Com.  Prop,  in  Fractions, 138 

Passive  Causes, 141 

To  find  Principal,  in  Interest,. ...  142 

To  find  the  Time, 143 

To  find  the  Rate, 143 

Causes  of  Capacity, 145 

To  find  the  Side  of  a  cubical 
figure  when  two  of  the  sides 

are  given, 145 

Relative  Contents  of  Hollow 

bodies, 147 

Contents  and  Sides  of  Cribs,.  .148 
Contents  and  Sides  of  Boxes, 

etc.,  in  bushels  and  gallons,.  149 
Size  and  relative  Weight  of  Me- 
tallic bodies, 150 

Novelty  in  Contraction, 152 

Com. Prop,  by  single  statement,153 

Rule  for  Com.  Proportion, 154 

Simple  Inverse  Proportion,  ....  155 

Theory  of  statement, 156 

Insufficiency  of  com'n  method,.  159 
Inv.  Proportion  in  Fractions, . .  159 

Calculations  in  Machinery, 162 

To  find  number  of  Revolutions, 162 

To  find  Size  of  Wheel, 163 

To  find  the  number  of  Teeth  or 

Diameter  of  Wheel, 164 

Concluding    remarks    on    the 

Proportions, 166 

Rule  for  Inverse  Proportion,  . .  167 
Conjoined    Proportion,  or  Chain 

rule, 168 

Theory  of, 168 

Exchange  of  Moneys, 170 

Rule  for  Conjoined  Prop 171 

Fellowship  Simple, 174 

Fellowship  in  Fractions, 176 

General  Average, 177 

Fellowship  Compound, 179 

Rule  for  the  Fellowships    and 

General  Average, 181 

Barter  and  Commerc'l  Exchange,]82 

Barter  by  Reduction, 184 

Origin  of  State  Currencies,.  ..185 
Combinations  of  statements,.. 186 
Rule    for  Barter,    Commercial 
Exchange,  and  Reduction,..  187 

Duties,  and  Tare  and  Tret, 188 

Law  relating  to, 189 

Definitions  in  Tare  and  Tret,.  .189 


PAGE. 

Specific  Duties, 190 

Ad  Valorem  Dut  ies, 191 

Law  relating  to, 192 

Tare  and  Tret  proper, 193 

Combination  of  statements,  .  .195 

Rule  for  Duties  &  Tare  &  Tret,196 

Commercial  Exchange  proper,  ..197 

By  combination, 197 

Rule  for  Commerc'l  Exchange^" 

Decimal  Fractions, \  _ 

Theory  of  Decimals, 200  ) 

Addition  of  Decimals, 203 

Subtraction  of  Decimals, 204 

Multiplication  of  Decimals, . . .  204 

By  Contraction, 205 

Division  of  Decimals, 208 

By  Contraction, 209 

To  reduce  Decimal  to  common 

Fractions, .210 

To  reduce  com.  to  Dec.  Frao.211 
To  reduce  Denominate  numbers 

to  Decimals, 212 

Mensuration,  or  Practical  Geom- 
etry,  213 

Theory  and  general  remarks,.  .213 

Measurem't  of  Wood  and  Bark,  216 

Combination  of  statements,  .  .217 

Measurement  of  Lumber, 219 

Combination  of  statements,  .  .220 

Cubic  Measurement, 221 

By  Combination, '221 

Masonry, 223 

Plasterers'  and  Pavers' Work,.. 224 

By  Combination, 225 

Carpenters'  and  Joiners'  Work,.  .226 

Cribs,  Boxes  and  Bodies, 227 

Wine,  Beer  and  Dry  Measure,227 

Combination  of  rules, 228 

To  find  the  Side  of  a  Crib,  Body, 

Box,  etc.— Rule, ...228 

Tonnage  of  Vessels, 229 

Weight  of  water  per  cubic  foot 

and  inch, 229 

Cylinders,  Spherical  and  Conical 

inches,  Sea-water,  etc 229 

U.  States  and  English  Standards 

of  Liquid  Measure, 230 

U.  States  and  English  Standards 

of  Dry  Measure, 230 

The  Winchester  Bushel, 230 

The  Connecticut  Bushel, 230 

Government  rule  for  Tonnage. 230 

Carpenters'  Rule, 232 

Superficial  Geometry, 232 

Variety  of  Figures,  Definitions 

and  Derivations, 233 

To  find  the  Contents  of  a  Rectan- 
gle,  235 


viii 


INDEX. 


PAGE. 

Contents  of  a  Parallelogram  and 

Rhomboid, 235 

Triangles, 236 

Polygons, 238 

The  Circle, 239 

Definitions  and  Derivations,  ....240 

Quadrature  of  the  Circle, 241 

History  and  difficulty  of, 241 

Inscribed    and     Circumscribed 

Polygons, 242 

Ratio    between    the  Diameter 

and  Circumference, 242 

To  find  the  Circumference  of  a 

Circle, 243 

Area  of  the  Circle, 244 

The  Ellipse, 247 

Circumference  of, 247 

Area  of,.. 247 

Contents  of  a    Square    Inscribed 

in  a  Circle, 248 

Solidity  of  a  Log  when  Squared,. 249 

Side  of  an  Inscribed  Square, 249 

To  find  the  largest  Square  that  a 

Round  Stick  will  make, 259 

The  Side  of  a  Square  given  to 
find  the  Diameter  of  a  Cir- 
cumscribed Circle, .250 

To  find  the  Diameter  of  a  Circle 
circumscribed  about  a  Square, 
or  how  large  a  Round  stick 
must  be  to  make  a  Square  of 

given  Side, 25 1 

To  find  the  Side  of  a  Square  of 
Area  equal  to  that  of  a  given 

Circle, 251 

To  find  Diameter  and  Circumfer- 
ence of  same, . 251 

Measurement  of  Cisterns, 252 

Square  and  Circular, 253 

Conical  and  Pyramidal, 254 

To  find  the  Side  or  Diameter  of 

a  Cistern, 255 

Table  of  Cisterns, 256 

Solid  bodies;  definitions  and  de- 
rivations,   257 

The  Cylinder, 258 

Superficial  Contents, 258 

Solid  Contents, 259 

Contents  of  Boilers, 260 

Table  of  Contents  of, 269. 


PAGE. 

Boilers  and  hollow  Cylinders,. 261 

Air  Pressure, 261 

Cones  and  Pyramids, 262 

Convex  Surface, 262 

Solidity  of, 262 

Frustum  of, 863 

The  Sphere  or  Globe, 264 

Surface  of,... .....264 

Solidity  of, 264 

Globe  and  Cylinder  compared,265 

The  Spheroid, 265 

Solidity  of, 265 

Proportions  among  Lines,  Areas 

and  Solidities, 265 

To     ascertain    the     Weight   of 

Globes, 265 

Weight  of    iron  Cylinders    and 

Globes  given,  26^ 

Gauging  Casks, 2Rt 

Mechanical  Powers, 26? 

The  Lever, 26^ 

The  Wheel  and  Axle, 269 

The   Inclined   Plane, 269 

The  Wedge, 270 

The  Screw, 270 

Square  Root, 271 

Currency, 274 

Customhouse  value  of  Foreign 

Coins, 274 

Table  of   Moneys   of  Account 

at  Congress   valuation,  ....275 
Jewish    Standard   Weights   and 

Measures, 277 

Jewish  Standard  of  Money,  and 

Table  of  Value, 278 

Table  of  Areas  of  Valves,  Circles, 

etc., , 279 

Table  of  weight  of  Square  Rolled 

Iron, 280 

Table  of  weight  of  Round  Rolled 

Iron, 281 

Table  of  weight  of  different  bo- 
dies of  Cast  Iron, 281 

Table  of  weight  of  Flat  Bar  Iron,282 
Table  ofweig.  of  Cast  Iron  Pipes 
and  Cylinders  from  1  to  30  in. 

diameter, 284 

Fac  simile  Coin  Plates  represent- 
ing the  coins  used  by  all  na- 
tions,  285  to  316 


KAINE  Y"  S 


IMPROVED    ABACUS. 


CANCELATION. 

ALL  arithmetical  computations  are  effected 
by  increase  and  decrease,  which  depend  in  their 
relations,  on  the  converse  operations  of  Multi- 
plication and  Division.  The  latter  are  but 
.  abbreviated  methods  of  adding  and  subtracting. 
Increase  and  decrease  are  the  results  of  the 
relative  difference  between  different  numbers 
and  quantities  of  the  same  thing :  hence  their 
result  depends,  in  all  reckoning,  on  the  great 
principles  of  Ratio  and  Proportion.  Therefore, 
by  Proportion,  as  the  rationale  of  statement,  and 
Multiplication  and  Division  as  the  mechanical 
media  of  reducing  such  statements  to  their 
results,  we  have  in  a  few  words,  an  epitome 
of  all  arithmetic. 

As  by  this,  Multiplication  and  Division  are 
presented  as  the  leading  operations  of  reckon- 
ing, we  may  profitably  spend  some  little  time, 
in  ascertaining  a  more  expeditious  method  of 
determining  products  and  quotients,  than  by 
the  old,  tedious,  and  circumlocutoiy  formulae 
of  the  books. 

When  7  is  multiplied  by  3,  the  product  is  21 : 
this  product  divided  by  another  3,  gives  7  again; 


6  RAINEY'S   IMPROVED   ABACUS. 

the  7  is  not  changed  by  the  Multiplication  and 
Division :  it  may,  therefore,  be  inferred,  that, 
when  any  number  is  both  multiplied  and  divided 
by  any  other  number,  the  former  remains  un- 
changed. Hence,  such  multipliers  and  divisors 
may  be  dropped,  as  useless,  and  the  numbers 
canceled.  If  the  7  be  multiplied  by  5,  and  the 
product  divided  by  10,  the  result  will  be  3i,  or 
i  of  the  7 ;  because  the  5  has  only  one  half 
the  capacity  in  elevating,  that  the  10  has  in 
depressing :  consequently,  the  7  is  affected 
twice  as  much  by  Division  as  by  Multiplication. 
We  may  therefore  divide  10  by  5,  and  place 
the  quotient  2,  on  the  side  of  the  10,  which 
shows  the  relation  between  these  numbers; 
the  one,  increasing  by  multiplication,  on  the 
right ;  the  other,  decreasing  by  division,  on  the 
left.  Again : 

Two  numbers,  as  12  and  16,  sustain  to  each 
other  a  relation,  that  may  be  expressed  by 
smaller  numbers.  They  may  be  reduced  to 
such  smaller  numbers,  by  extracting  a  factor 
or  figure  which  has  been  instrumental  in  pro- 
ducing the  numbers  in  each  case.  Thus,  12  is 
composed  of  4  times  3,  while  16  is  composed 
of  4  times  4.  Here  the  same  factor  which  has 
been  used  in  making  each  number,  the  4,  may 
be  extracted  in  each  case,  leaving  3  in  the  12, 
and  4  in  the  16  :  which  shows  that  \  f  are  equal 
to  | ;  or,  that  i-f  make  J.  In  this  case,  as  in 
all  other  cases  of  factors,  the  number  sup- 
posed, must  not  be  written  down;  and  must 
be  contained  without  a  remainder,  in  some 
number  on  both  the  right  and  left  of  the  line. 
Such  numbers  on  the  two  sides  of  the  line,  as 


CANCELING.  7 

are  divided  by  the  supposed  factor,  may  be 
canceled,  and  the  other  constituent  factor  must 
be  set  on  the  side  of  the  canceled  number, 
from  which  taken.* 

When  there  are  numbers  on  the  right  and 
left,  terminated  by  ciphers,  these  ciphers  may 
be  stricken  off  in  equal  numbers,  as  so  many 
factors  of  10.  In  |£  we  cancel  the  two  ciphers, 
and  leave  f  ;  which  is  equivalent  to  extracting 
10  in  each  case.  No  cipher  can  be  canceled 
which  has  a  significant  digit  on  its  right ;  for  its 
value  is  qualified  by  such  digit.  Numbers  on 
the  left  of  ciphers  may  be  canceled  with  other 
numbers  on  the  opposite  side  of  the  line  ;  as 
such  numbers  are  but  factors  co-operating  with 
the  ciphers  on  the  right,  to  constitute  their 
sum.  Ciphers  thus  isolated  indicate  10.  From 
these  considerations  we  conclude,  that,  to  can- 
cel numbers,  after  they  have  been  arranged 
on  the  two  sides  of  the  vertical  line, 

1st.  Cancel  all  equal  numbers  on  the  two  sides 
of  the  line  : 

2d.   Cancel  ciphers  in  equal  numbers  : 

3d.  Divide,  leaving  no  remainder,  from  one 
side  of  the  line  into  the  other,  and,  vice  versa, 
placing  the  quotient  on  the  side  of  the  larger  : 

4th.  Extract  all  possible  factors  from  any  two 
numbers  occupying  different  sides  of  the  line, 
and  leave  the  other  constituents  of  such  numbers, 
on  the  side  of  each. 

A  few  examples  in  the  multiplication  of 
fractions  are  given,  merely  to  illustrate  the 

*  Cancel,  is  from  the  French  cancelkr,  which  signifies 
literally,  to  cross  a  writing. 


8         RAINEY'S  IMPROVED  ABACUS. 

application  of  the  foregoing  directions.  Multi- 
ply |  of  f  of  TV  of  2,  by  «  of  Lf  of  li  of  fi  of  5. 
Here,  as  in  the  multiplication  of  all  fractions, 
we  place  all  the  Numerators  on  the  right,  and  all 
the  Denominators  on  the  left.  We  have  not 
sufficient  space  to  give  the  theory  of  these 
statements,  in  this  little  treatise,  which,  it  is 
designed  to  devote  more  particularly  to  opera- 
tions in  practical  business ;  leaving  all  the  work 
preparatory  to  this,  to  lectures  at  the  black- 
board, or  to  the  elementary  works  of  others.* 

The  following  remarks  may  be  proper  just 
here  : 

The  upper  part  of  a  fraction  is  called  the 
Numerator ;  and  the  lower  part,  the  Denomina- 
tor. The  Denominator,  from  de,  concerning, 
and  nomen,  a  name,  shows  the  name  of  the 
fraction,  or  the  number  of  parts  into  which  the 
unit  or  whole  thing,  is  divided.  The  Nu- 
merator, from  numerus,  number,  shows  the 
number  of  parts,  of  the  size  indicated  by  the 
Denominator,  taken.  A  whole  number  is  con- 
sidered a  numerator,  whose  denominator  would 
be  1.  Mixed  numbers,  such  as  4i,  3|,  &c., 
before  placed  on  the  line,  must  be  reduced  to 
improper  fractions.  This  is  done  by  multiply- 
ing the  whole  number  by  the  denominator  of  the 
appended  fraction,  and  adding  in  the  numerator. 
The  denominator  of  the  number  is  again  used, 
as  the  denominator  of  the  improper  fraction. 
Thus,  in  4i,  twice  4  make  83  and  1  makes  |. 
In  3^,  five  times  3  are  15,  and  1  is  '/.  Hence, 

*  Day  and  Thomson's  Practical  and  Higher  Arithme- 
tics. 


MULTIPLICATION   OF   FRACTIONS. 


I 


8— £ 

t— 


8Ans. 


9  and    16  are  the  numerators,  although  they 
are  larger  respectively,  than  their  den  ominators. 

Fours  equal :  5  into  10,  twice  ; 
this  2  equals  2  opposite  :  8  into 
24,  3  times,  while  3  times  3  on 
the  right  make  9,  which  goes 
into  18  on  the  left  twice  :  now 
twice  6  on  the  left  equals  12  on 
the  right :  5  into  35  seven  times, 
and  7  into  14  twice;  this  2  into 
16  eight  times,  on  the  left:  we 
have  remaining  on  the  left  8, 
and  on  the  right  5 ;  making 
f ,  Ans. 

After  canceling  as  far  as  practicable  on  the 
two  sides  of  the  line,  multiply  continuously 
together  all  the  terms  on  the  right,  for  a  new 
numerator,  and  all  the  terms  on  the  left,  for  a 
new  denominator.  If  nothing  remains  on  the 
right,  one  is  understood.  If  the  number  on  the 
right  be  smaller  than  the  number  on  the  left, 
the  answer  is  a  fraction ;  of  which,  in  all 
cases,  the  right  is  the  numerator  ;  but,  if  the 
number  on  the  left  be  smaller,  the  right  must 
be  divided  by  the  left :  in  such  case,  the 
answer  will  be  a  mixed  number. 

What  will  7i  Ibs.  iron  come  to  at  2|  cts.  per 


lb.?     Here    7i    make 


1_5 


and 


2 1  make    l-j 


We  place  the  numerators  on  the 
right,  and   the  denominators  on  the     ft 
left.     Five  into  15,  three  times  ;  and       1 18  cts. 
2  into   12  six  times  ;  while  six  times 
3,  the  only  remaining  numbers,  make  18  cts., 
the  answer. 


10  RAINEY'S  IMPROVED  ABACUS. 

Multiply,  -jf  of  a  yard  of  cloth,  by  }f  of  a 
dollar  per  yard,  thus, 

_  3         Here,  we  must  suppose  some 
£  —  3     factor,  before  the  numbers  can  be 


20 


20 


reduced.  Let  us  take  5,  which 
goes  into  25  five  times,  and 
into  15  three  times ;  again  :  4  into 
16  four  times,  and  into  12  three  times;  we 
have  5  and  4  as  factors,  neither  of  which  has 
been  placed  down ;  only  their  quotients.  Now 
3  times  3  on  the  right,  and  4  times  5  on  the 
left,  make  ^  of  a  dollar. 

What  will  2f  yards  of  gambroon  come  to 
at  1.20  cents  per  yard? 

Eight  is  contained  in  120,  15 
times ;  and  this  number  multiplied 
by  21  on  the  same  side,  gives  $3,15 
cents  for  the  answer. 


#0-15 


3,15 


If  T85  of  a  farm  are  divided  among  4  heirs, 
how  much  will  each  get? 

In  this  instance  the  dividend,  T8j,  must  be 
placed  on  the  right  of  the  line,  and  4,  the  di- 
visor, on  the  left;  thus, 

A  fractional  number  occupies  the 


15 


*— 2 


right  or  left  of  the  line,  when  its  nu- 
merator is  on  the  right  or  left.  Let 
15j  2  the  numerator  be  located  first;  then, 
15  I  the  denominator  is  merely  placed  op- 
posite. If  I  direct  the  pupil  to  place 
T9T  on  the  left  of  the  line,  he  must  place  the  9 
on  the  left  only,  and  the  17  on  the  right.  The 
numerator  always  indicates  the  locality  and 
value  of  the  fraction. 

Divide  J  of  f  of  20,  by  TV  of  f  of  '/  of  \ j 
of  4i  of  v- 


COMPLEX    FRACTIONS. 


11 


The  numerators  of  the  divi- 
dend are  placed  on  the  right,  and 
those  of  the  divisor  on  the  left, 
with  all  the  denominators  oppo- 
site their  respective  numerators. 

Four  into  20  five  times,  and  5 
equals  5  on  the  left;  9  into  18 
twice,  and  twice  2  on  the  right 
make  4,  which  goes  into  12  on 
the  left  3  times;  5  into  25  five 
times,  and  this  5  again  into  10 
on  the  right  twice ;  8  into  40  five 


7 
A 

3-/W 
t-tt 

5-40 


105 


16 


105 


times ;  3  into  9  three  times ;  3  equals  3 ;  2  and  8 
remain  on  the  right,  and  5,  3,  and  7  on  the  left, 
which  multiplied  separately  give  TW 

From  the  foregoing  we  may  deduce  the  fol- 
lowing directions : 

To  multiply  fractions ;  place  all  of  the  numera- 
tors, both  of  the  multiplicand  and  multiplier,  on 
the  right  of  the  vertical  line,  and  all  the  denomina- 
tors on  the  left. 

To  divide  fractions;  place  the  numerators  of 
the  dividend  on  the  right,  and  those  of  the  divisor 
on  the  left,  with  the  respective  denominators  of 
each  opposite  their  numerators. 


COMPLEX    FRACTIONS. 

(It  is  not  designed  in  this  short  treatise  to  devote  ranch  space  to  either 
the  theory  or  practice  of  fractions  ;  as  it  is  believed  that  there  are  very 
many  elementary  works  accessible,  which  do  entire  justice  to  this  depart- 
ment of  numbers.  We  shall  merely  introduce  a  page  or  two  on  complex 
fractions,,  for  the  consideration  of  teachers,  that  we  may  present  a  short 
and  simple  method  of  using  them,  which  is  not  found  in  the  books.) 

Division  of  ordinary  fractions  leads  to  the 
consideration  of  those  that  are  complex.     The 


12  RAINEY'S  IMPROVED  ABACUS. 

doctrine  may  be  advanced,  that  to  increase  the 
terms  of  a  fraction,  so  as  not  to  change  its  value, 
multiply  both  the  numerator  and  denominator  by 
the  same  number. 

If  |  be  multiplied  by  2,  it  is  made  f  .     Now, 

3_L 
if  instead  of  f  ,  the  fraction  were-j-,  we  might 

double  the  numerator  3i,  by  multiplying  by 
the  special  denominator  2,  and  adding  in  the 
special  numerator  1  ;  thus  making  J  of  the 
numerator. 

But  if  this  numerator  is  thus  increased  by 
the  2,  the  denominator  should  be  likewise  : 
hence  the  4  is  multiplied  by  the  same  2,  mak- 
ing 8,  by  which  we  have  t,  which  is  equivalent 

31 

to  -p.      Both   terms   of   the   fraction    are   in- 
4 

creased  by  the  same  number,  the  denominator 
of  the  fraction  annexed  to  the  numerator. 
After  reducing  the  numerator,  it  may  be  ex- 

7 

pressed  thus  :  T  —  -r,  showing  that  the  two  de- 
" 


nominators  have  been  thrown  together,  and 
may,  consequently,  be  combined  in  multipli- 
cation. 

It  appears  thus,  that  if  we  would  reduce 
complex  to  simple  fractions,  and  at  the  same 
time  to  the  lowest  term,  we  should  multiply 
each  term  of  the  fraction  by  the  denominator  or  de- 
nominators of  the  fraction  or  fractions  annexed 
to  the  numerator  or  denominator,  or  both,  adding 
in  at  each  separate  multiplication,  the  given  nume- 
rator or  numerators.  For  example,  in  the  frac- 


COMPLEX    FRACTIONS.  13 


tion  o^,we  multiply  first  by  the  denominator 

of  the  numerator,  2.  Twice  3  are  6,  and  one, 
the  numerator,  added,  makes  7,  or  J:  now 
we  multiply  the  denominator  2|,  by  the 
same  2; — twice  2  are  4,  and  twice  i  are  f, 
making  4|.  This  4f  must  be  placed  under 

the  7,  thus,  j^.     We  have   now  reduced    the 

complex  fraction  in  the  numerator  by  multi- 
plying it  and  the  whole  denominator  below,  by 
the  denominator  2  above,  and  proceed  to  that 
of  the  denominator.  Taking  the  new  fraction 
4|  to  work  on,  we  say  3  times  4  are  12  and  2 
are  14,  for  a  new  denominator,  and  3  times  7 
are  21,  for  a  new  numerator;  thus,  f-J.  This 
reduced  to  its  lowest  term,  by  dividing  the  nu- 
merator by  the  denominator,  gives  14  for  the 
answer. 

It  is  seen,  therefore,  that  in  each  case,  it  is 
necessary  to  reduce  both  the  numerator  and 
the  denominator  to  a  mixed  number.  After 
this  is  done,  knowing  that  the  denominator 
of  a  fraction  is  the  divisor,  we  may  deduce  the 
following  rule: 

To  reduce  complex  to  simple  fractions,  of  the 
lowest  term :  Reduce  both  terms  to  an  improper 
fraction,  and  place  the  numerator  of  the  numera- 
tor on  the  right,  and  the  numerator  of  the  denomi- 
nator on  the  left,  with  their  respective  denominators 
opposite. 

Let  us  reduce  in  this  way  ^.  The  nume- 
rator 34  is  first  reduced  to  J  and  placed, 
2 


14  RAINEY'S  IMPROVED  ABACUS. 

the  numerator  on  the  right  of  the  line,  and 
the  denominator  on  the  left;  thus, 

Next  reduce  the  denominator  2J  to  an 
improper  fraction,  and  divide  by  it,  plac- 
ing the  7,  or  numerator,  as  in  other 
cases  of  division,  on  the  left.  Seven  equals  7 ; 
and  2  is  contained  in  3  one  and  a  half  times, 
which  is  the  answer,  as  above. 

By  this  process,  the  complex  is  not  only  re- 
duced to  a  simple  fraction,  but  to  the  lowest 
term  of  that  fraction ;  all  of  the  factors  being 
excluded  in  the  canceling:  while  it  is  plain 
and  intelligible,  and  at  once  proves  itself  ne- 
cessarily correct  to  all  who  take  the  trouble  to 
know  that  the  numerator  of  a  fraction  is  the 
dividend,  and  the  denominator  the  divisor. 

In  the  same  manner  that  other  fractions  are 
multiplied,  we  multiply  these,  by  placing  on 
the  right  the  numerators,  both  in  the  multipli- 
cand and  multiplier.  Hence, 

To  multiply  complex  fractions  and  reduce  them 
to  their  lowest  term :  Place  the  numerators  of  the 
numerators,  both  of  the  multiplicand  and  mul- 
tiplier, on  the  right;  the  numerators  of  the  denomi- 
nators on  the  left ;  and  all  respective  denominators 
opposite. 

Multiply  ^  by  j|.     In    the    first    place   8j. 

make  \5 ,  which  are  placed  on  the  right,  while 
4,  the  denominator,  is  placed  with  the  3  on  the 
left.  Under  this  are  placed  the  4|,  or  y ,  the 
other  numerator;  and  f,  the  denominator,  is 
placed  on  the  left;  thus, 


COMPLEX    FRACTIONS. 


15 


Four  times  6  on  the  left  equal  24  on 
the  right.     The  answer  is  8J. 


25 

f 


•  ~3 

1  6. 

Again:   Multiply  fby~.     Thus,  J  is  divided 

4"  9" 

by  f ,  and  this  is  multiplied  by  f  and  divided 
by*. 

Three  is  contained  in  6  twice,  and  $  f 

twice  4  equals  8  on  the  left.     Seven  £  £ 

equals  7,  while  5  is  contained  in   9  f  $ — fi 

If  times.  5I9 


Division  of  complex  fractions  may  be  per- 
formed as  in  the  division  of  other  fractions, 
by  inverting  or  placing  on  the  left  of  the  line 
the  numerator  of  the  divisor.  Hence, 

To  divide  one  complex  fraction  by  another'. 
Place  the  dividend  on  the  right  and  the  divisor  on 
the  left. 

4—       2— 
Divide  ~  by  ^|.     Here  we  place  4|  on 

right,  and  7i  on  the  left ;  and  dividing  by  % 
place  it  on  the  left,  while  its  denominator 
is  placed  opposite. 

4] 

It  appears  that  all  of  the  numbers    frfr 
equal:  hence,  one  is  understood  on  the 
right  for  the  quotient. 


the 


.11 

It  is  foreign  to  our  purpose  to  fill  this  little 
volume  with  examples  for  the  reader  to  solve ; 
nence,  after  pointing  out  clearly  the  principles 


16 

governing  statements  and  solutions,  we  shall 
give  him  the  privilege  of  working  such  ques- 
tions as  his  experience  and  business  may  sug- 
gest ;  a  few  examples,  however,  will  be  intro- 
duced, to  familiarize  him  with  canceling  and 
using  figures  by  this  system. 

Multiply  3J  Ibs.  cheese,  by  8J  cents  per  Ib. 

Multiply  J  of  T\  of  20,  by  y  of  f  of  T8T  of 
JL  of  f£  of  10i  of  180. 

3-^  2— 

Multiply  -1  of  40  by  —-,  and    divide    by    f 

of  TV 

Divide  i  by  i,  and  multiply  it  by  4  of  20. 

The  examples  above  are  deemed  sufficient 
to  enable  the  learner  to  proceed  with  ease, 
after  the  questions  are  stated,  without  a  teacher. 
If  the  numbers  on  the  two  sides  of  the  line 
are  such  as  cannot  be  canceled,  they  must  be 
multiplied  and  divided,  and  the  result  will  be 
the  same.  The  vertical  line  will  be  explained 
in  Simple  Proportion. 


SIMPLE  INTEREST.* 

$5-  The  following  method  of  casting  interest  is  designed  for  every  rate 
percent.  Many  of  the  questions  wrought  under  this  head,  can  be  wrought 
with  less  than  half  the  number  of  figures  here  required,  by  the  short  6  per 
cent.,  and  other  methods  which  will  follow. 

SIMPLE  INTEREST  is  an  allowance  made  for 
the  use  of  money,  and  is  different  in  different 
countries.  It  is  supposed  that,  as  in  one  year 
the  products  of  the  soil,  on  which  all  other 
profits  are  primarily  based,  yield  their  regular 

*  I  am  indebted  to  the  Rev.  William  McGookin,  of  Ohio,  for  many  impor- 
tant suggestions  on  simple  interest. 


THEORY  OF   INTEREST.  17 

increase,  so  it  is  reasonable  that  a  borrower 
of  money  should  be  required  to  refund  the 
amount  borrowed,  annually.  Hence,  the  unit 
of  time  for  which  money  is  lent,  is  one  year ; 
called  per  annum  or  by  the  year,  from  the  Latin 
words,  per — by,  and  annum — a  year.  The 
charge  of  a  specific  price  or  bonus  for  the  use 
of  money,  requires  that  a  sum  be  established, 
as  a  general  sum,  on  which,  in  one  year,  the 
specified,  legal  interest  shall  be  charged;  so 
that  a  larger  sum  would  receive  proportionally 
more,  and  a  smaller  sum  proportionally  less 
interest ;  while  likewise  a  greater  or  less  length 
of  time  than  one  year,  would  make  the  inter- 
est proportionally  more  or  less.  This  specified 
sum  is  100;  a  number  chosen  because  easily 
used  in  dividing ;  being  the  round  product  of 
two  decimates.  The  100  is  called  per  centum, 
from  per — by,  and  centum,  a  hundred.  Hence 
per  centum  per  annum  means,  by  the  hundred, 
by  the  year.  Cent,  is  a  contraction  of  centum. 
Most  of  the  States  of  our  Union  have  estab- 
lished 6  per  cent,  per  annum,  as  the  legal  rate 
of  interest.  Now,  it  is  manifest  that  if  per 
centum  or  100,  in  per  annum  or  1  year,  that 
is,  if  100  dollars  in  one  year,  gain  6  dollars 
interest,  a  larger  or  smaller  number  of  dollars 
will  gain  more  or  less  than  6  dollars,  in  one 
year ;  while  likewise,  $100  in  a  greater  or  less 
time  than  one  year,  would  gain  more  or  less 
than  6  dollars  interest.  If  $100  in  one  year  gain 
6  interest,  200  would  gain  twice  6,  or  12;  and 
if  200  in  one  year  gain  12,  in  two  years  it  would 
gain  twice  twelve ;  or  in  one  half  of  .a  year 
the  half  of  12,  or  6.  So  we  perceive  that  the 


18  RAINEY'S   IMPROVED  ABACUS. 

interest  on  a  sum  of  money  for  a  given  period 
of  time  depends  on  the  relation  that  such  sum 
bears  to  100,  as  well  as  the  relation  of  the 
time  to  1  year.  These  relations  are  properly 
ascertained  by  Compound  Proportion ;  but  we 
will  here  present  them  in  the  form  of  two 
connected  simple  proportions.  What  is  the 
interest  on  $50  for  six  months,  at  6  per  cent.  ? 
that  is,  if  $100  in  12  months  gain  $6  interest, 
what  interest  will  $50  gain  in  6  months  ? 
Now  we  place  the  demand,  $50  and  6  months, 
on  the  right  of  a  line,  and  the  terms  of  the 
same  name,  100  dollars  and  12  months,  as  the 
supposition,  opposite  these,  on  the  left.  When 
we  place  dollars  opposite  dollars  thus,  it  is  to 
get  the  ratio  between  the  two  numbers  of 
dollars :  the  same  is  the  case  with  the  months, 
which  are  placed  opposite.  The  ratio  between 
100  and  50  is  i  ;  and  between  12  months  and 
6  months  \ ;  so  that  the  two  ratios  multiplied 
together,  that  is  \  times  i,  make  \ ;  hence,  the 
$6  interest  multiplied  by  this  i,  is  reduced  to 
li,  which  is  the  result ;  thus, 

The  ratio  of  these  numbers 
compared,  is  obtained  as  by  a 
fulcrum  and  scale.  Hence,  50  is 
contained  in  100  twice,  and  6  in 
12  twice,  and  this  2  in  6  three 
times,  and  2  on  the  left  in  3  on  the  right  li 
times,  which  is  \\  dollars,  the  answer.  Suppose 
the  time  above,  instesd  of  being  6  months,  were 
6  days ;  then  we  cc  aid  not  place  one  year 
opposite  it  in  the  form  of  12  months  ;  but  only 
in  the  form  of  360  days  :  for,  to  obtain  the 
ratio  of  time  that  6  days  bear  to  a  year,  we 


£— 3 


PRACTICE   OF   INTEREST.  19 

must  compare  days  with  days,  not  with  months : 
hence  if  the  time  were  6  days,  we  would  place 
360  days,  which  make  a  year,  opposite;  or 
which  is  the  same  thing,  30  days  opposite,  to 
make  a  month,  and  12  months  to  make  a  year : 
these  two  numbers  make  360,  the  number  of 
days  in  an  interest  year.  Let  us  find  the 
interest  for  6  days ;  thus, 


2— ,100  £0 


20 


Jl^ 

20 


We  must  place  on  the  left  op- 
posite the  specified  time,  1  year, 
12  months,  or  360  days,  just  as 
the  time  specified,  may  be  in  years, 
months  or  days.  Here  again,  50 
into  100  twice  :  6  times  6  on  the 
right  equal  36  on  the  left ;  leaving 
the  cipher  or  10  to  be  multiplied  into  the  2, 
making  the  denominator  20.  One  being  un- 
derstood on  the  right,  the  answer  is  ^V  °f a 
dollar;  equal  to  5  cents. 

We  see  then,  that  100  and  the  time,  either 
in  years,  months  or  days,  go  to  the  left ;  while 
we  place  the  sum  on  which  the  interest  is 
to  be  obtained,  the  time,  and  the  rate,  all  on 
the  right. 

Now  if  the  rate  be  6,  or  10,  or  any  other  per 
cent.,  it  is  so  many  hundredths  ;  100  being  the 
denominator,  while  6  or  10,  &c.,  is  the  nume- 
rator. This  denominator  is  composed  of  two 
decimals.  Decimal  is  from  the  Latin  word 
decent,  which  means  ten  ;  hence,  the  decimal  is 
the  tenth  part  of  a  unit.  Two  of  these  deci- 
mals, or  the  tenth  times  one  tenth,  make  one 
one-hundredth :  hence,  any  two  decimal  factors, 
express  so  many  hundredths.  It  is  necessary 
in  expressing  a  decimal  fraction,  that  there 


20  RAINEY'S   IMPROVED   ABACUS. 

be  one  figure  less  in  the  numerator  than  in 
the  denominator:  and  the  denominator  100, 
containing  3  figures,  we  always  consider  that 
there  are  but  two  in  the  numerator  or  rate. 
Then,  if  we  consider  the  rate  per  cent,  two 
decimals,  every  thing  multiplied  into  it  must  be 
made  100  times  smaller  :  that  is,  if  dollars  be 
multiplied  by  the  rate,  they  become  hundredths 
of  dollars,  or  cents  :  and  if  cents  be  thus  multi- 
plied by  the  rate,  or  two  decimals,  they  become 
hundredths  of  cents.  Now,  in  the  first  question 
wrought,  let  us  make  this  rate  two  decimal 
factors,  by  dropping  the  100  at  the  left.  It  is 
unnecessary  here  to  place  a  cipher  at  the  left 
of  the  6,  to  show  that  the  rate  is  composed  of 
two  decimals  :  this  will  be  understood. 

Certainly,  when  we  multiply  the 
$50  by  this  rate  per  cent.,  6,  it  is 
made  100  times  smaller  than  dol- 
lars, and  becomes  cents ;  so  that 
the  answer  is  150  cents.  We 
will  therefore  cut  off  two  figures  at  the  right 
of  the  result  for  cents.  Suppose  again,  we 
consider  this  50  cents,  instead  of  50  dollars: 
then  the  cents  being  multiplied  by  the  rate, 
become  hundredths  of  cents ;  so  that  in  the 
answer,  we  cut  off  two  for  hundredths  of  cents, 
two  more,  if  we  have  them,  for  cents,  while 
the  remaining  figures  at  the  left,  are  dollars. 
But  having  only  3  figures,  the  answer  is  1  cent 
and  50  hundredths;  or  li  cents.  Let  us  get 
the  interest  on  $60,  for  317  days,  at  6  per  cent. 
We  call  the  $60  here,  as  we  call  the  sum  in 
all  other  cases,  the  PRINCIPAL — the  317  days, 
the  TIME,  and  the  6  per  cent,  per  annum,  the 


50 


INTEREST   IN   FRACTIONS.  21 

RATE.     We  place  these  numbers  on  the  right, 
thus, 

and  as   the  time  is  in  days,  we  00 

place  30   and  12  opposite,  or  360,  £0317 

which   is    the  same  thing  ;     and     #    A#  r 
again  dispense  with  the  100.     If  $|3,17 

we  dispense  with  this  100  on  the  | 
left,  the  answer  will  be  100  times  smaller  than 
the  principal;  and  this  being  dollars,  the  an- 
swer will  be  cents.  Cyphers  equal  :  6  into 
12  twice  :  and  twice  3  on  the  left,  equal  6  on 
the  right.  We  have  317  left,  and  conclude, 
that  the  answer  is  3  dollars  and  17  cents.  If 
the  principal  were  60  cents,  the  answer  would 
be  3  cents  and  17  hundredths.  From  this  we 
conclude,  that,  When  the  Principal  is  dollars, 
the  answer  is  cents;  and  when  the  principal  is 
cents  ,  the  answer  is  hundredths  of  cents. 

What  is  the  interest  on  $80,  for  9  months, 
at  7  per  cent.  ?  Here,  we  place  the  P.  T.  &  R. 
as  before,  on  the  right  ,  and  place  12  only,  on 
the  left;  because  the  time  is  in  months,  or  T9^ 
of  a  year. 


We  use  the  factor  4,  which 
goes  into  12  three  times,  and  into 
8  twice  :  3  into  9  three  times  : 
now  3X2X7X10  make  420,  or 
$4  and  20  cents.  Were  this 


$0  —  2 
X  —  fi-&  ft  —  3 


$|4,20 


time  9  days,  we  would  place  30  with  12  on 
the  left.  Were  the  principal  80  cents,  the 
answer  would  be  four  cents  and  T\\. 

What  is  the  interest  on  37  1  cents  for  18 
days  at  74  per  cent.  ?  The  principal  and  rate 
being  mixed  numbers,  must  be  reduced  to  im- 
proper fractions  ;  and  the  numerators  placed 


22  RAINEY'S   IMPROVED   ABACUS. 

on  the  right,  with  their  denominators  on  the 
left.  The  numerator  of  a  fraction  in  all  cases, 
occupies  the  same  place  that  otherwise  the  whole 
number  would :  while  the  denominator  is  invaria- 
bly placed  opposite.  Hence  again,  the  time  being 
days,  we  place  30  and  12  on  the  left.  The 
principal  is  cents  ;  \5  cents  ;  hence,  the  answer 
will  be  hundredths  ;  and  as  such,  we  will  strike 
off  two  numbers  for  hundredths,  and  two  for 
cents. 

2  75  j       Fifteen  into  30,  twice,  and 

2 — #0  /l$_ 0 — 3  2  into  18,  9  times  :  The  fac- 
4 — £&  tor  3  into  9  three,  and  into 

12,  four  times;  4X2X2  are 
16  on  the  left,  and  3X75  on 
the  right,  are  225 ;  which  di- 
vided by  1 6  gives  1 4  Tl¥ .  The 
answer  is  no  dollars,  no  cents,  and  14  TJ¥  hun- 
dredths cents.  Such  examples  as  this  are 
scarcely  of  any  practical  value,  and  only  show 
the  full  extent  of  the  theory  and  practice  of 
interest  by  this  system. 

What  is  the  interest  on  $600,60  cents,  for  3J 
years,  at  4i  per  cent.?  We  make  the  3  J  years, 
y ,  and  place  the  10  on  the  right,  and  3  on  the 
left,  and  divide  by  nothing,  except  the  denom- 
inator. All  that  we  divide  by  the  numbers 
12,  and  12  and  30  for,  is  to  reduce  the  time  to 
years ;  hence,  when  the  time  is  already  in 
years,  division  by  any  number  becomes  unne- 
cessary, except  by  such  denominators,  as  from 
mixed  numbers,  may  fall  on  the  left ;  which 
is  the  case  with  3  and  2  in  this  example. 
Here,  the  4i  make  |  per  cent. 


INTEREST   FOR  BROKEN  TIME.  23 


Two  into  10  five,  and  3  into 
9  three  times:  now  3X5X600,60 
make  900900.  We  cut  off  two  for 
hundredths,  and  two  for  cents : 


600,60 

£  40—5 
'  — 3 


$|90,09,00 


hence,  the    answer  $90,09    cents, 
and  no  hundredths.     This  answer  is  in  hun- 
dredths, because  the  principal  is  in  cents. 

What  is  the  interest  on  $600,  for  3  years,  6 
months  and  20  days,  at  6  per  cent.  ?  Here  it 
is  necessary  to  reduce  all  the  years  to  months, 
and  add  in  the  given  months  ;  and  likewise  re- 
duce the  days  to  the  fractional  part  of  a  month, 
and  add  such  fraction  to  the  months.  In  three 
years  there  are  36  months,  and  6  more  added, 
make  42  months.  Now,  20  days  are  f  %-  of  a 
month,  which,  canceling  the  twro  ciphers,  makes 
§.  The  time,  therefore,  is  42f  months,  which 
make  -f-  months.  We  place  this  128  on  the 
right,  and  3  on  the  left.  The  time  now  being 
in  months,  we  divide  by  12  only. 

Six  into  12  twice,  and  twice  000 

3    on   the  left,  equal    6  on  the  %  128 

right:    we    consequently    draw     # — £40 

down   the   128,  and   annex  the  f|128,00 

two  ciphers,  making  the  answer 
$128,00.  Suppose  the  time  had  been  1  year, 
1  month,  and  10  days.  One  year  and  1  month 
make  13  months :  10  days  are  1^  or  i  of  a 
month :  consequently  the  time  is  13 J,  or  \° 
months.  Here,  40  should  go  to  the  right,  and  3 
to  the  left,  with  12.  Again  :  Suppose  the  time 
6  months  and  15  days  :  These  15  days  are  i.f 
or  \  month ;  so  that  the  time  is  64,  or  l-/ 
months.  Here,  again,  12  should  be  placed  on 
the  left.  Suppose  the  time  2  years,  9  months 


24  RAINEY'S  IMPROVED   ABACUS. 

and  25  days:  the  25  days  make  thus,  |f, equal 
to  |  of  a  month  ;  and  two  years  and  9  months, 
make  33  months,  which,  with  the  f  annexed, 
is  33 1  or  2  f 3  months.  Suppose  the  time  27 
days  :  this  would  be  f  J,  or  T\  of  a  month,  to  be 
annexed  to  all  the  months.  Were  it  28  days, 
it  would  be  ||,  or  1  j  of  a  month.  Nine  days 
would  be  JL,  or  T%.  of  a  month :  so  would  8 
days  be  -J^,  or  T4j  of  a  month.  Suppose  the 
time  were  three  months  and  29  days.  Most 
business  men  would  call  this  30  days  :  but  to  be 
accurate,  we  would  multiply  the  months  by  30 
and  add  in  the  29.  Thus,  the  whole  time  would 
be  reduced  to  119  days  :  and  we  would  conse- 
quently divide  by  30  and  12.  When  the  days 
make  a  number  that  cannot  be  reduced  to  the 
fraction  of  a  month,  to  secure  entire  accuracy, 
the  years  must  be  reduced  to  months,  and  all 
the  given  months  added  in  ;  then,  these  months 
must  be  reduced  to  days,  and  the  given  days 
added  in. 

What  is  the  interest  on  $50  for  1  year,  3 
months  and  5  days,  at  6  per  cent.?  One  year 
and  3  months  make  15  months:  5  days  are  1 
of  a  month:  hence  151,  or  y  months,  is  the 
time.  We  divide  by  12  only. 

Sixes  equal:  we  have  12  on  the 
91          left,  and  on  the  right  50X91  which 
makes    4550.     This   divided   by    12 
gives  for  answer  $3,791  cents. 

From  the  foregoing  principles  and 
operations,  we  are  justified  in  ma- 


12|4550 


king  the  following 


RULE  FOR  ALL  INTEREST.  25 


SUMMARY  OF  DIRECTIONS, 

For  working  Interest  of  any  conceivable  Prin- 
cipal, Time,  and  Rate. 

Place  the  Principal,  Time,  and  Rate,  on  the 
right  of  the  vertical  line  ;  and  if  the  time  is  days, 
place  30  and  12  on  the  left:  if  the  time  is  months, 
place  12  only,  on  the  left:  and  if  the  time  is 
years,  place  nothing  on  the  left. 

If  the  Principal,  Time,  or  Rate  is  a  mixed 
number,  reduce  it  to  an  improper  fraction,  and 
place  the  numerator  on  the  right,  with  the  denom- 
inator on  the  left. 

When  the  Principal  is  dollars,  the  answer  is 
cents:  in  such*' case,  two  figures  must  be  cut  off 
for  cents :  when  the  Principal  is  cents,  the  an- 
swer is  hundred  ths  of  cents:  here,  cut  off  two  fig- 
ures, commencing  at  the  right,  for  hundredths, 
two  more  for  cents,  and  the  remainder  at  the  left 
is  dollars.  The  figures  thus  cut  off  for  cents, 
hundredths,  tyc.,  must  be  whole  numbers;  while 
any  existing  fraction  will  be  only  a  fractional 
part  of  such  cents  or  hundredths. 

When  the  time  is  months  and  days,  or  years, 
months  and  days,  reduce  the  years  to  months,  and 
add  in  all  the  given  months :  then  reduce  the  days 
to  the  fractional  part  of  a  month,  and  annex  this 
fraction  to  the  whole  number  of  months:  reduce 
all  to  an  improper  fraction,  and  place  the  numerator 
on  the  right,  and  \he  denominator  on  the  left.  In 
such  case,  divide  by  12  only.  If  the  time  cannot 
be  reduced  to  the  fractional  part  of  a  month,  re- 
duce tfie  whole  time,  years,  months  and.  days,  to 
days,  and  divide  by  30  and  12. 


26  RAINEY'S  IMPROVED  ABACUS. 

If  4ht  time  is  years  and  months,  reduce  the 
months  to  the  fractional  part  of  a  year :  add  to 
the  years :  reduce  all  to  an  improper  fraction,  and 
divide  by  the  denominator  only. 

If  the  answer  to  a  question  be  $80,  20  cents 
and  38|f  hundredths,  and  it  were  written  thus, 
80.20.38^1,  it  would  be  wrong  to  cut  off  eith- 
er the  18  or  29  by  itself,  or  both  together,  for 
the  denomination  of  hundredths;  for  they  make 
only  the  i|  part  of  one  one  hundredth  part  of 
a  cent.  Hence,  to  strike  off  cents  or  hun- 
dredths of  cents,  place  the  separatrix  between 
integral  numbers  only. 

The  use  of  360  days  to  the  year,  may  be  by 
some  thought  singular;  but  it  grows  out  of  the 
standard  commercial  usage,  3€  days  to  the 
month,  and  12  months  to  the  year.  The  busi- 
ness year,  the  civilized  world  over,  is  called 
360  days.  If,  however,  any  wish  to  use  the 
365,  they  can  easily  do  so  by  substituting  365 
on  the  left,  for  30  and  12.  The  difference  in 
the  result  is  only  T*¥  part.  In  banks,  the  in- 
terest is  always  reckoned  for  three  days  more 
than  the  time  specified  by  the  borrower,  which 
are  called  days  of  grace,  or  days  given  the  bor- 
rower to  allow  for  any  accidents  or  exigencies 
which  may  prevent  the  money  being  funded 
at  the  close  of  the  discounting  period.  Grace 
means  gift.  Really  the  three  days  are  not  days 
of  grace ;  for  interest  is  reckoned  on  them  as 
part  of  the  general  discounting  time.  Banks, 
too,  charge  more  than  the  legal  rate  of  inter- 
est ;  on  what  principle  of  ethics,  however,  I 
have  never  been  able  to  learn.  If  the  rate  be 
6  per  cent.,  the  note  for  one  year,  is  given  for 


INTEREST  FOR  MONTHS  AND  YEARS.  27 

$100;  and  the  interest  on  $100,  which  is  $6,  is 
deducted  from  the  money  when  issued  to  the 
borrower;  so  that  he  gets  only  $94.  Now 
here,  he  pays  $6,  not  for  the  use  of  100,  which 
would  be  equitable,  at  6  per  cent.,  but  for  94. 
The  interest  on  $94,  the  sum  that  the  borrow- 
er receives,  at  6  per  cent,  will  not  be  $6;  so 
that-  he  loses  clear  the  difference  between  the 
$6,  and  the  sum  of  interest  that  94  would  gain. 
This  difference  is  quite  important  in  heavy 
transactions. 

It  is  frequently  impossible  to  cancel  in  ques- 
tions of  interest ;  when  this  is  the  case,  all  the 
numbers  on  the  right  must  be  multiplied  to- 
gether for  a  dividend,  and  all  on  the  left  for  a 
divisor  :  after  which  the  former  must  be  divi- 
ded by  the  latter.  Some  would  ask,  "  what 
benefit  in  working  interest  in  this  way,  if  at 
times,  it  is  necessary  to  multiply  and  divide,  as 
in  the  old  system  ?"  It  is  because  all  ques- 
tions, whatever  be  the  principal,  time  or  rate, 
can  be  wrought  by  this  one,  simple  rule,  with- 
out a  separate  rule  for  every  varying  per  cent.; 
and  which  rule  itself,  is  based  on  a  principle 
so  remote,  as  seldom  to  be  seen  by  the  ordi- 
nary arithmetician  :  because  the  statement 
can  be  easily  made  and  understood  by  any  or- 
dinary mind ;  because  the  work  is  unique  and 
systematic;  and  because  in  most  cases  the  work 
can  be  greatly  abbreviated  by 
canceling.  We  give  but  two 

other     examples,     and     these    3 ^ 

without   the  work.     What   is          ~~3'65000 

the  interest  on  $800,  for  2  years, 

8  months  and  15  days,  at  10  per  |.  b' 


65 
10 


28  RAINEY'S  IMPROVED  ABACUS. 

cent.?     Here  then,  the  time  makes  -V-  months. 


30 
12 


10,000 
90 


What  is  the  interest  on  $10,000 
for  90  days,  at  i  of  1  per  cent.? 
This  time  may  be  placed  on  the  line 
as  90  days  3  months,  or  i  of  a 
year.  In  the  first  case,  we  would 

divide  by  30  and  12  ;  in  the  second  by  12  only, 

and  in  the  last  by  nothing. 


6,25 


INTEREST  AT  SIX  PER  CENT. 

Interest  at  6  per  cent,  has  long  since  been 
reckoned  by  dividing  by  60,  when  the' time  was 
days,  or  when  months,  multiplying  the  princi- 
pal by  half  their  number.  This  process  is 
very  easy,  when  the  number  of  months  is 
even,  and  the  half  can  be  found  without  ma- 
king it  necessary  to  multiply  the  principal  by 
a  mixed  number  ;  but  when  the  time  is  an  odd 
number,  or  has  years,  months  and  days  ;  or 
when  the  principal  is  a  mixed  number,  it  is 
difficult,  by  the  process  ordinarily  pursued,  to 
use  the  fractions,  and  ascertain  the  precise 
result.  Hence,  the  time  is  generally  made  too 
great  or  too  little,  and  many  fractions  are 
thrown  away  ;  whereas  by  the  use  of  the  ver- 
tical line,  and  the  consequent  advantage  of 
placing  the  numerators  and  denominators  on 
its  two  sides,  this  difficulty  is  entirely  obvia- 
ted ;  so  that  if  the  numbers  on  the  two  sides 
cannot  be  canceled,  they  can  at  least  be  mul- 
tiplied and  divided,  as  by  the  old  method.  By 
this  method,  therefore,  a  clear  gain  is  made  of 


INTEREST  AT  SIX  PER  CENT.  29 

all  numbers  that  can  be  canceled,  as  well  as 
greater  ease  and  perspicuity  in  the  statement. 
We  find  by  the  first  method  presented  in  this 
work,  that  the  interest  on  one  dollar,  for  60 
days,  at  6  per  cent.,  is  one  cent.  It  is  likewise 
the  same  for  2  months,  thus  : 


|1  cent.  |1  cent. 

In  the  former  case,  the  time  being  60  days, 
30  and  12  are  used  on  the  left;  in  the  latter, 
the  time  being  2  months,  12  only  is  used.  The 
result  is  the  same. 

If  one  dollar,  as  above,  give  1  cent  interest 
in  60  days,  it  will  in  6,  which  is  the  tenth  part 
of  60  days,  give  the  tenth  part  of  a  cent,  or 
one  mill.  The  fact  is  therefore  established, 
that 

One  dollar  in  6  days,  at  6  per  cent.,  will  gain 
ONE  MILL  interest ;  and  ONE  DOLLAR  in  2  months 
at  6  per  cent.,  will  gain  ONE  CENT  interest.  Hence, 
we  are  justified  in  making  the  following 
statement  in  Proportion:  If  1  dollar  in  6  days 
on  the  left,  give  one  mill  interest,  last  on  the 
right,  how  many  mills  will  any  other  number 
of  dollars  and  days  give  on  the  right?  Or, 
if  one  dollar,  in  2  months  on  the  left,  give  one 
cent  interest,  last  on  the  right,  how  many 
cents  will  any  other  number  of  dollars  and 
months  give,  on  the  right  ? 

What  is  the  interest  on  $40  for  240  days? 
Here,  the  principal  and  time  are  placed  for  a 


30  RAINEY'S  IMPROVED  ABACUS. 

demand,  on  the  right,  1  dollar  and  6  days,  for 
the  same  name  on  the  left,  and  1  mill,  last  on 
the  right,  for  the  denomination  of  the  answer: 
Thus.  The  answer  is  consequently, 
mills ;  hence,  one  figure  must  be  cut 
|  j  off  for  mills,  and  all  at  the  left  of  it 

jj-gQ-Q-     are  cents  and  dollars.     The  answer 
is  one  dollar,  60  cents  and  no  mills. 
The  ones  are  placed  on  the  two  sides  of 
the  line,  merely  to  indicate  the  proportion  in 
the    statement.     They    are    unnecessary    in 
practice,  and  may  be  dropped  in  the  statement 
and  calculation. 

What  is  the  interest  on  20  cents 


01794 


3|7940 


for  794  days  ?  Two  is  contained  in 
six  three  times,  which  we  can  di- 
vide by,  no  farther,  and  consequent- 
ly bring  down  on  the  left.  The 
794  are  multiplied  by  10,  by  merely  appending 
the  cipher.  In  this  case,  the  principal  is 
cents ;  hence,  the  answer  is  1000  times  smaller 
than  if  it  were  dollars,  and  is  consequently 
thousandths  of  cents.  We  strike  off  one  fig- 
ure for  thousandths,  two  more  for  hundredths 
of  cents,  and  the  remaining  figure  is  cents  ; 
hence  the  answer,  2  cents,  94  hundredths,  and 
6§  thousandths.  The  business  man  every- 
where, would  call  this  3  cents.  By  this  it  is 
seen  that,  when  the  principal  is  dollars,  the  an- 
swer is  mills,  and  when  the  principal  is  cents,  the 
answer  is  thousandths  of  cents. 

It  is  quite  preferable  that  the  answers 
should  be  in  dollars,  cents,  and  hundredths  of 
cents.  To  effect  this,  when  the  time  is  days, 
and  we  divide  by  6,  as  in  the  foregoing,  we 


INTEREST  AT  SIX  PER  CENT.  31 

may  cut  off  and  throw  away  at  the  right  of 
the  answer,  one  figure,  for  mills  or  thousandths 
of  cents,  as  useless;  and  the  answer  will  be 
cents  or  hundredths,  according  to  the  denom- 
ination of  the  principal.  It  must  be  remem- 
bered, that  this  figure  is  thrown  away  at  the 
right  of  the  answer,  only  when  the  time  is 


MO  -  3401 

What  is  the  in-    ^g_3i     10203 
terest  on  680,20  cts.     -%p  --  qrnrzrtt    A 
for  93  days?  $10,54,310  Ans. 

Here,  the  factor  3,  is  contained  in  6  twice, 
and  in  93  thirty-  one  times  ;  this  2  in  6802  is 
contained  3401  times.  We  multiply  the  lat- 
ter number  by  31,  by  using  the  3  only,  not  set- 
ting it  down,  but  placing  its  product  one  move 
to  the  left  of  the  unit's  place,  and  adding  the 
two  numbers.  To  this  sum  we  annex  the  ci- 
pher. One  figure,  the  cipher,  is  thrown  off, 
which  leaves  the  answer  10  dollars,  54  cents, 
and  31  hundredths.  The  long  method  of  mul- 
tiplying by  31  would  be  quite  as  easy  for 
most  persons  as  that  use'd  above.  Or  the  two 
original  numbers  680,20  and  93  could  be  mul- 
tiplied, and  their  product  divided  by  the  6,  pro- 
ducing the  same  result. 

It  may  be  remarked  here,  that  whenever  it 
is  necessary  to  multiply  by  the  numbers  21, 
31,  41,  51,  61,  71,  81,  91,  the  left  figure  only, 
may  be  multiplied  by,  and  its  product  removed 
one  figure  to  the  left  of  the  unit's  place,  and 
the  two  numbers  added.  Likewise  to  multi- 
ply by  13,  14,  15,  16,  17,  18  and  19,  use  the 
right  hand  figure  only,  placing  the  product  one 
move  to  the  right,  and  add  as  before.  In 


32 


RAINEY'S  IMPROVED  ABACUS. 


11,750 


87,5 


neither  of  the  cases  do  we  write  the  number 
multiplied  by. 

What  is  the  interest  on  87^  cents,  for  120 
days  ?  In  this  instance  we  may  place  the 
principal  on  the  line  as  a  mixed 
number  -J-,  as  in  the  annexed  ex- 
ample, or  we  may  make  the  i  cents, 
5  decimals,  and  place  it  down  as  87,5, 
thus, 

The  result  will  be  the  same,  ex- 
cept that,  in  the  latter  case,  one 
figure  must  be  cut  off  for  the  deci- 
1 1,75, 00  mals;  then  the  figures  remaining 
may  be  treated  as  usually.  In  the  first  ques- 
tion, 6  times  2  equal  12  on  the  right;  in  the 
2d,  6  into  12  twice,  and  twice  87,5  are  1750, 
which  with  the  10  annexed,  becomes  17500. 
The  result  is  the  same,  after  cutting  off  the 
decimal  in  the  latter  case,  and  afterwards  one 
figure  in  each  case  for  the  thousandths.  Hence, 
the  answer,  one  cent  and  75  hundredths. 

What  is  the  interest  on  287,37i  cents,  for 
80  days  ?  We  give  the  i  cent  here  the  deci- 
mal expression,  .5,  and  will  throw  away  one 
figure  in  the  answer  for  it. 


Here  multiply  by  8,  annex 
the  cipher,  and  divide  by  6.  The 
answer  is  3  dollars,  83  cents, 
16  hundredths,  &c.,  &c. 


What  is  the  interest  on  $200, 
for  15  days?     Ans.  Fifty  cents. 


Jrt) 


22990000 


3,83,16,66^ 


£00 


,500 


SIX  PER   CENT.  INTEREST. 


33 


What  is  the  interest  on  $360,  for     Jgf  " 

97  days  ? 

$|5,820 

We  now  come  to  questions  in  which  the 
time  is  months,  or  months  and  days,  or  years, 
months  and  days.  In  this  case  2  is  used  on 
the  left,  because  in  2  months  $1  gains  1  cent 
interest;  the  answer  will  be  in  cents  and  hun- 
dredths, without  throwing  off'  one  figure. 

What  is  the  interest  on  $200,  for  7 
months  ?  Here  we  place  principal  and 
time  on  the  right,  and  2  on  the  left ; 
the  answer  is  $7,  and  no  cents. 

Interest  on  $387,20  cents,  for  10 
months?  Two  into  10  five  times, 
and  five  times  387,20  are  19  dol- 
lars 36  cents,  and  no  hundredths, 
answer 


387,20 


19,36,OC 


What  is  the  interest  on  $47,  for  5 
months?  In  this  instance,  we  multi- 
ply and  divide  only. 


47 


|235 


What  is  the  interest  on  $480, 
93|  cents,  for  1  month  ?  Three- 
fourths  are  made  .75,  in  the  form 
of  two  decimals  :  hence,  in  the 


480,93,75 
1 


2,40,46,87^ 


answer,  two  figures  are  cut  off  for  decimals, 
two  for  hundredths,  and  two  for  cents. 


What  is  the  interest  on  12J  cents, 
for  20    months  ?     This    principal   is 
reduced  to   halves :    answer,   1  cent 
and  25  hundredths ;  or  li  cents. 
3 


25 


34 


RAINEY'S  IMPROVED  ABACUS. 


13 


$|2,60 

$  ^00—2 
11 


$|22,00 


What  is  the  interest  on  $80,  for  6 
months  and  15  days  ?  15  days  being 
i  of  a  month,  the  time  is  6i,  or  \3- 
months.  Hence  the  statement. 

What  is  the  interest  on  $1200,  for 

3   months  and  20  days  ?     Twenty 

20 
days  are  |E,  equal  to  f  of  a  month  : 


hence  the  time  3f,  or  -U-  months.  In  these 
instances,  the  denominators  are  placed  on  the 
left  with  the  2  months. 

What  is  the  interest  on  $1000,  for  1 1 
months  and  12  days?  Twelve  days 
are  1J>  equal  to  |  of  a  month:  the 

^iK1?  nn  3  o  *      JL  5 

b7>UL  time,  consequently,  is  11|,  or  -y- 
months.  Here,  5  times  2  on  the  left,  equal  10, 
or  cipher  on  the  right.  The  answer  is  $57,00. 
What  is  the  interest  on  $1500,  for  three 
years,  8  months  and  25  days  ?  Three  years 
and  8  months  make  44  months  :  and  25  days 
are  J  of  a  month,  which  makes  the  time  44 1 
months.  This  reduced  to  an  improper  fraction, 
in  £|-2.  months  :  thus, 

Twice  six  on  the  left,  are  12, 
which  goes  into  1500,  one  hun- 
I  dred  and  twenty  five  times  :  and 
$|336,25         I  269X125=$336,25,  the  answer. 
What   is  the  interest  on    80  dollars,  for   4 
years,  10  months  and  9  days  ? 
Four    years    and    10   months, 
make   58   months,   and  9  days 
make    y\   of  a  month,   which 


,4^00 ^5 

269 


40583 

H|  23,32  Ans. 


are  58  T\,  equal  to  -5T\3-  months. 

In  cases  where  the  time  is  an  even  number 
of   years,  it  is   only   necessary   to    multiply 


RULE  FOR  SIX  PER  CENT.  INTEREST.  85 

together  principal,  time  and  rate,  and  cut  off  2 
or  4  figures  for  cents,  as  indicated  by  the  de- 
nomination of  the  principal. 


SUMMARY  OF  DIRECTIONS. 

When  the  Time  is  DAYS,  place  the  Principal  and 
Time  on  the  right,  and  6  on  the  left.  If  the 
Principal  is  dollars,  the  answer  is  MILLS;  here, 
cut  off  one  figure  at  the  right,  for  mills,  and  two 
more  for  cents :  if  the  Principal  is  cents,  the  an- 
swer* is  thousandths  of  cents;  here,  cut  off  one 
figure  for  thousandths,  two  for  hundredths,  and 
two  more  for  cents.  Or,  cut  off  one  figure  at  the 
right  in  cither  case,  and  the  answer  will  be  cents, 
or  hundredths,  according  to  the  denomination  of 
the  principal. 

When  the  Time  is  MONTHS,  place  Principal 
and  Time  on  the  right,  and  2  on  the  left.  If 
the  Principal  is  dollars,  the  answer  is  cents  ;  rf 
cents,  the  answer  is  hundredths  of  cents. 

When  the  Time  is  YEARS,  place  Principal, 
Time  and  Rate  on  the  right,  and  multiply  contin- 
uously. The  answer  is  cents  or  hundredths,  ac- 
cording to  Hie  denomination  of  the  principal. 

When  the  Time  is  years  and  months,  or  years, 
months  and  days,  reduce  the  years  to  months,  and 
add  all  the  given  months:  then  reduce  i/te  days  to 
the  fractional  part  of  a  month,  if  practicable ; 
annex  this  fraction  to  the  months :  reduce  all  to 
an  improper  fraction,  and  place  the  numerator 
en  the  right,  and  denominator  on  the  left. 


36  RAINEY'S  IMPROVED  ABACUS. 

If  the  days  cannot  be  reduced  to  the  fractional 
part  of  a  month,  reduce  the  whole  time,  years, 
months,  and  days,  or  months  and  days,  to  days, 
and  divide  by  6,  as  in  other  cases. 

When  the  principal  is  a  mixed  number,  reduce 
it  to  an  improper  fraction,  and  place  the  numera- 
tor on  ike  right,  and  denominator  on  the  left :  Or, 
express  the  fraction  in  decimals,  and  cut  off  as 
many  figures  on  the  right  of  the  answer  for  deci- 
mals, as  indicated  by  the  number  of  decimals  in  the 
principal. 

INTEREST    AT    SEVEN    AND    EIGHT    PER    CENT. 

The  first  method  given  in  this  book  for  reck- 
oning at  any  rate  per  cent.,  is  quite  applica- 
ble to  7  and  8  per  cent.  Yet  we  may  use 
a  method  which  requires  fewer  figures,  in  con- 
nection with  the  6  per  cent,  method  just  noticed. 

In  states  where  the  rate  of  interest  is  7  and 
8  per  cent.,  it  will  be  found  very  convenient  to 
ascertain  the  interest  first  at  6  per  cent.,  by 
the  short  rule  given,  and  then  add  |  more  for 
7  per  cent.,  or  i  for  8  per  cent.  One  per  cent, 
is  the  i  of  6  per  cent.:  hence  we  may  divide 
the  interest  at  6  per  cent,  by  6,  and  thus  get  £, 
which  will  be  added  to  the  former  interest. 
So  two  per  cent,  over  6,  making  8,  is  £  of  6: 
hence  the  interest  at  6  may  be  divided  by  3, 
and  the  quotient  added  to  the  former  interest. 

When  the  rate  is  12  per  cent.,  instead  of  di- 
viding by  6  as  in  6  per  cent,  calculations,  3 
only  may  be  placed  on  the  left  of  the  line,  and 
the  answer  divided  off  as  in  6  per  cent,  calcu- 
lations ;  because  at  this  rate  per  cent,  one  cfol- 


SEVEN    AND    EIGHT    PER   CENT.    INTEREST.         37 


lar  in  three  days  gives  one  mill  interest.     A  few 
examples  are  given  for  illustration. 

What  is  the  interest  on  $350  for  160  days  at 
7  per  cent.?     We  work  as  in  6  percent.;  thus, 


350 

3—0  i00— 8 


28000 


9,33,3 


$  10,88,8i 


Placing  6  on  the  left,  at  6  per 
cent,  the  answer  would  be 
$9,33i  cents.  We  now  place  6 
at  the  left  of  this  answer,  which 
is  equivalent  to  multiplying  the 
answer  by  |,  not  using  the  nu- 
merator on  the  right,  and  divide 
the  first  answer,  setting  the  quo- 
tient beneath.  This  added  gives  for  the  an- 
swer, $10,88  cents  and  8i  mills. 

Again :  What  is  the  interest  on  $20  for  8 
months  at  7  per  cent.? 

Here,  2  is  used  on  the  left,  the  time 
being  months,  and  the  answer  80  cents 
is  obtained.  One-sixth  or  13J  added 
gives  93J  cents.  One  advantage  in 
this  method  is  that  the  numbers  to  be 
divided  by,  on  the  left, 


20 


80 


,931 


on    the    leit,    are    never 

large,  and  the  fraction  lost  by  the  last  division, 
is  of  no  practical  importance. 

What  is  the  interest  on  $397,20  for  40  days 


at  7  per  cent.? 

The  answer  in  this  instance 
is  $3  and  nearly  9  cents. 


What  is  the  interest  on 
8  per  cent? 


0  ^,20—662 
40 


264800 
441331 


$  3,08,93,3i 
for  371  days  at 


38 


RAINEY'S  IMPROVED  ABACUS. 


6 


^0—3 
371 


11130 
3710 

;  14,84,0 

Again : 
days  at  8 


In  this  example,  after  finding  the 
interest  at,  6  per  cent.,  it  is  divided 
by  3,  which  is  equivalent  to  multi- 
plying by  i,  and  the  quotient  is  ad- 
ded, making  14  dollars  and  84  cents, 
answer. 

What  is  the  interest  on  $399  for  20 
per  cent.? 


1330 


The  answer  is  one  dollar,  77 
cents,  and  3J  mills. 


What  is  the  interest  on  $487,20  for  4  months 
at  8  per  cent.? 

487,20 

^  ^ — 2  The  time  being  months,  we  di- 

97440  vide  first  by  2,  and  afterward,  the 

32480  answer  thus  obtained  by  3. 


$  12,99,20 

What  is  the  interest  on  $1500  for  211  days 
at  12  per  cent.? 
ii£00  _  5  |       We  here  divide  by  3  only,  be- 


#2 


211 


cause,  as  said  above,  1  dollar  in  3 
days  at  12  per  cent,  gives  1  mill  in- 
$[105,50,0  terest.  Hence,  one  figure  is  cut 
off  in  the  answer  for  mills. 

When  the  time  is  months  and  the  rate  12 
per  cent.,  the  principal  and  time  may  simply 
be  multiplied  together;  because  1  dollar  in  1 
month  at  12  per  cent.,  gives  1  cent  interest.  Hence 
nothing  more  is  necessary  than  multiplication, 
as  1  placed  on  the  left  of  the  line  could  serve 


RATES    OF    INTEREST.  39 

no  other  purpose  than  to  show  the  nature  of 
the  statement  by  proportion. 

What  is  the  interest  on  $873  for  7  months 
at  12  per  cent.? 


The  answer  i*in  this,  as  in  all  other 
cases  of  months,  obtained  in  cents,  61 
dollars  and  11  cents. 


873 

_7 

1,11 


DIRECTIONS     FOR    SEVEN,     EIGHT,     AND     TWELVE    PER 
CENTUM. 

State  as  in  cases  of  6  per  cent.:  if  the  rate  is 
7  per  cent.,  divide  the  answer  by  6,  and  add  the 
quotient:  if  the  rate  is  8  per  cent.,  divide  the  an- 
swer by  3,  and  add  the  quotient. 

If  the  rate  is  12  per  cent.,  and  the  time  days, 
place  principal  and  time  on  the  right  and  3  on  the 
left:  cut  off  one  figure  at  the  right,  and  the  an- 
swer will  be  cents  or  hundredths  of  cents. 

When  the  time  is  in  months,  multiply  principal 
and  time  together,  and  the  answer  will  be  in  cents 
or  hundredth  of  cents. 

When  the  time  is  years,  multiply  principal, 
time,  and  rate  together;  and  the  answer  will  be 
cents,  or  hundredths  of  cents. 

RATE  AND  FORFEITURE  TABLE. 

RATES  OF  INTEREST  legalized  in  the  several  states  of  our  union,  and  in 
foreign  countries,  with  the  PENALTIES  for  USURY.* 

FTATES.  RATES.!  PENALTIES  FOR  USURY. 

Maine,  6  per  cent.  Forfeit  of  entire  debt. 

New  Hampshire,  6    "      "  Three  times  the  usury. 

Vermont,  6    "      '*  Usury  recoverable  with  costs. 

Massachusetts,      6    "      "  Three  times  the  usury  forfeited. 

*  On  all  dues  to  the  United  States,  6  per  cent,  is  charged,  ever;  where  the 
states  legalize  higher  rates. 

t  If  the  rate  is  not  mentioned  in  the  note,  interest  may  be  collected  at  the 
rate  established  by  the  law  s  of  the  state  in  which  the  transaction  occurs. 


40 


RAINEY  S    IMPROVED    ABACUS. 


RATE   AND    FORFEITURE    TABLE. 


STATES. 
R.  Island, 

RATES. 

6  per  cent. 

PENALTIES  FOR  USURY. 
Forfeit  of  interest  and  usury. 

New  York, 

7 

cc       cc 

cc           cc           cc       cc      ' 

New  Jersey, 

6 

cc       cc 

cc          cc          cc       cc 

Pennsylvania, 

6 

cc       cc 

cc          cc          cc^    cc 

Delaware, 

6 

cc       cc 

cc          cc          cc       cc 

Maryland, 

6 

cc       cc 

(1) 

Such  contracts  void. 

Virginia, 

6 

cc       cc 

Forfeit  twice  the  usury. 

North  Carolina, 

6 

cc       cc 

cc              cc                  cc 

South  Carolina, 

7 

cc       cc 

Forfeit  interest,  usury,  and  costs. 

Georgia, 

8 

cc       cc 

Forfeit  three  times  the  usury. 

Alabama, 

8 

cc       cc 

Forfeit  usury  and  interest. 

Mississippi, 
Louisiana, 

8 
5 

cc       cc 

(2) 
(3) 

Forfeit  of  usury  and  costs. 
All  such  void. 

Tennessee, 

6 

cc      cc 

cc       cc       cc 

Kentucky, 

6 

cc       cc 

Costs  and  usury  recoverable. 

Ohio,      ' 

6 

cc       cc 

All  such  void. 

Indiana, 

6 

cc        c 

Forfeit  of  twice  the  usury. 

Illinois, 

6 

cc       cc 

(4) 

Forfeit  interest  and  three  tirm 

BS  usury 

Missouri, 

6 

CC         Ct 

(5) 

Forfeit  interest  and  usury. 

Michigan, 
Arkansas, 
Florida, 
Wisconsin, 

7 
6 
8 

7 

cc       cc 

(6) 

(7) 

Forfeit  one-fourth  debt  and  us 
Whole  usury  forfeited. 
Forfeit  interest  and  usury. 
Forfeit  three  times  usury. 

mry 

Iowa, 

7 

cc      cc 

(8) 

cc             cc           cc           cc 

Texas, 

10 

cc       cc 

All  such  void. 

Dist.  Columbia, 

6 

cc       cc 

cc      cc        cc 

England, 

5 

cc       cc 

Forfeit  three  times  the  debt. 

France, 

5 

cc       cc 

Ireland, 

6 

cc      cc 

Canada, 

6 

cc       cc 

Nova  Scotia, 

6 

cc       cc 

W.  Indies, 

8 

cc       cc 

Constantinople, 

30 

cc       cc 

MAKING    AND    TRANSFERRING    NOTES. 

In  closing  the  article  on  interest,  it  may  be  well  to  give  a  few  practical 
directions  on  making  and  transferring  notes.  Much  litigation  arises  from 
inattention  to  the  following  requirements  of  law  : 

1.  A  promissory  note  is  an  instrument  of  writing  in  which  the  promisor 
or  maker  pledges  the  payment  of  money  or  property  to  a  second  person, 
at  or  before  a  specified  time,  in  consideration  of  equivalent  value  received. 

2.  The  sum  of  money  or  property  for  which  the  promisor  gives  the  note, 
is  called  the  "  face  of  the  note,"  and  after  being  expressed  in  figures  at  the 
beginning  of  the  note,  should  be  written  in  words  in  the  body  of  the  same 

3.  The  words  "  value  received  "  should  always  be  found  in  the  body  of 

(1)  Contracts  in  tobacco  may  be  as  high  as  8  per  cent. 

(2)  By  contract  as  high  as  10  per  cent. 

(3)  By  agreement  as  high  as  10;  bank  interest  6  per  cent. 

(4)  By  agreement  as  high  as  12. 

(5)  By  agreement  as  high  as  10. 

(6)  Any  rate  not  above  10  per  cent. 

(7)  By  contract  as  hig'h  as  12. 

(8)  By  agreement  as  high  as  12  per  cent. 


LAWS    REGULATING    NOTES.  41 

a  note,  as  the  laws  require  that  money  shall  be  paid  only  for  a  "  con- 
sideration "  or  equivalent.  Without  this,  notes  are  said  to  be  invalid  or 
worthless. 

4.  The  individual  who  makes  the  note  is  called  the  drawer  or  giver  ;  the 
person  to  whom  it  is  given  is  called  the  payee  ;  and  the  person  who  has  it 
in  legal  possession,  is  called  the  holder. 

5.  The  payee  of  a  note  may  sell  or  transfer  it  to  a  third  person,  if  it  be 
written  payable  "  to  order,"  or  "  to  bearer  ;"  and  this  third  person  or  holder 
may  sue  and  collect  as  if  he  had  been  the  original  payee.     Such  a  note  is 
called  negotiable,  because  it  can  be  traded  by  the  payee,  and  made  paya- 
ble to  such  person  as  he  may  order. 

6.  The  law  requires  the  holder  of  a  negotiable  note  to  indorse  it,  or  write 
his  name  on  the  back,  if  he  wish  to  sell  it ;  provided  that  such  note  be 
transferable,  or  "  payable  to  order."     If  the  holder  is  unable  to  collect  the 
note  of  the  drawer,  then  the  indotser  is  responsible,  and  can  be  made  to 
pay  it.     The  holder  of  a  note  which  is  made  payable  "to  bearer,"  can 
transfer  without  indorsing  it ;  and  is,  in  this  event,  not  liable  for  it.     In  the 
transfer  of  such  notes  there  is  said  to  be  no  recourse.     Thus,  if  a  bank- 
note is  not  indorsed  by  the  individual  who  pays  it,  he  is  no  longer  liable  to 
lose  it,  if  it  prove  worthless. 

7.  A  note  made   payable  to  a  particular  individual  without  the  words 
payable  "  to  order,"  or  "to  bearer,"  cannot  be  negotiated  or  traded  to  an- 
other ;  nor  can  another  individual,  except  in  the  name  of  the  payee,  col- 
lect it. 

8.  A  note  specifying  no  time  for  payment,  must  be  paid  on  demand. 
Such  are  generally  called  "  due-bills." 

9.  Conventional  usage,  and  in  some  instances  law,  has  established,  that 
a  note  shall  not  be  collected  until  three  days  after  it  is  due  ;  and  interest 
is  calculated  on  these  three  days,  as  on  the  rest  of  the  time.     A  note  be- 
coming due  on  Sunday,  with  these  three  days  included,  should  be  paid  on 
Saturday.     Three  days  thus  given,  is  to   allow  for  all  exigencies  ;  and  are 
called  "  days  of  grace,"  because  they  are  given  gratuitously.    Grace  means 
gift. 

10.  Maturity  of  a  note  is  the  time  specified  for  its  payment.    If  a  note 
is  not  paid  at  maturity  that  has  been  transferred,  the  holder  must  legally 
notify  the  indorser  of  the  fact,  or  the  indorser  is  released  from  his  liability. 

11.  Interest  cannot  be  charged  on   a  note  paid  at   maturity,  without  it 
has  been  specified  ;    the  words,    "  with  interest,"    being  inserted.      But 
when  a  note  not  containing  these  words,  is  not  paid  until   after  maturity, 
the  rate  of  interest  legal  in  the  state  can  be  charged  from  maturity. 

12.  Notes  bearing  interest  without  the  rate  being  inserted,  bear  the  legal 
interest  of  the  state  in  which  drawn.     Any  agreement  for  a  rate  of  inter- 
est less  than  the  legal  interest,  must  ba  specified,  or  legal  interest  can  be 
collected. 

13.  Notes  for  any  commodity  of  merchandise  cannot  be  negotiated  ;  and 
if  payment  is  not  made  at  the  time  specified,  the  holder  may  recover  the 
amount  in  money.     In  such  cases  the  three  days  grace  are  not  allowed. 

14.  When  two  or  more  persons  give  a  note  conjointly,  the  payee  or 
holder  may  collect  it  of  either  or  any  of  them. 

PAYMENT  ON  NOTES. — To  compute  interest  on  notes,  we  must  ascertain 
the  time  which  elapses  between  the  period  when  interest  commences,  and 


42  RAINEY'S  IMPROVED  ABACUS. 

that  on  which  the  payment  is  made,  by  subtracting  the  former  from  the  lat- 
ter date,*  which  is  done  thus: 

What  is  the  interest  on  a  note  for  $500,  dated  May  20,  1843,  and  paid 
January  19,  1845  ? 

After  arranging  the  former  time  tinder 
the '  latter  thus,  if  the  number  of  days  in 
the  lower  line  is  larger  than  that  in  the 


yrs.  mos.  days. 
1845  "  1  "  19 
1843  "  5  "  20 

Time,  1  "  7  "  29 


upper,  30  days  must  be  added  to  the  upper 
line,  and  the  subtraction  made  from  the 
whole  number  above,  and  the  remainder 
set  under  the  days.  One  is  carried  to  the 


act  uuuci   me  uuys.      wiie  is  uarncu  LU    me 

lower  line  of  months.  If  this  number  of 
months  is  larger  than  that  above,  12  must  be  added  above  and  the  subtrac- 
tion continued  as  before.  It  will  be  observed  here,  that  the  months  are 
placed  down  according  to  the  order  they  occupy  in  the  year.  May  is  the 
5th  month;  hence  we  use  5  as  the  number  ;  so  is  1  used  lor  January,  it  be- 
ing the  first  month. 


PARTIAL    PAYMENTS. 

Below,  we  give  the  Ohio  rule  for  casting  the 
interest  on  partial  payments,  which  is  the 
method  used  in  Indiana,  Kentucky,  and  most 
of  the  states  of  the  union. 

The  rule,  and  the  calculation  to  illustrate 
its  application,  are  extracted  from  Swan's 
Treatise. 

"  The  Ohio  rule  for  calculating  partial  pay- 
ments, is  as  follows :  Where  payments  exceed- 
ing the  interest  are  made  after  the  debt  is  due  : 
In  such  case  interest  should  be  calculated  on 
the  debt  up  to  the  time  of  payment,  and  the 
principal  and  interest  then  added  together, 
and  the  payment  subtracted  from  the  total. 
Subsequent  interest  should  be  computed  on 
the  balance  of  principal  thus  found  to  be  due. 

"Where  the  payment  is  less  than  the  inter- 
est due,  the  surplus  of  interest  must  not  be 
added  to  the  principal ;  but  interest  continues 
on  the  former  principal,  the  same  as  if  no 

*  The  day  on  which  a  note  is  dated,  and  that  on  which  it  becomes  due, 
•hould  not  both  be  reckoned.    The  former  is  excluded  among  business  men. 


PARTIAL  PAYMENTS.  43 

payment  had  been  made,  until  the  period 
when  the  payments  added  together,  exceed  the 
interest  due;  and  then  the  surplus  of  pay- 
ments is  to  be  applied  towards  discharging  the 
interest.  For  instance,  upon  a  note  for  $100, 
payable  in  one  year  with  interest,  if  a  pay- 
ment of  10  dollars  is  made  at  the  end  of  two 
years,  and  10  dollars  at  the  end  of  four  years, 
and  19  dollars  at  the  expiration  of  six  years ; 
here  interest  on  the  whole  amount  of  the  note 
should  be  calculated  up  to  the  time  of  the 
payment  of  the  19  dollars,  and  then  the  sever- 
al payments  should  be  added  together,  and 
deducted  from  the  amount  of  all  the  principal 
and  interest;  the  balance  would  be  the  amount 
due,  and  upon  which  interest  should  be  after- 
wards computed. 

The  following  calculations  will  illustrate 
the  rule  in  the  text: — A.,  by  his  note,  dated 
Jan.  1st,  1840,  promises  to  pay  to  B.  1000  dol- 
lars, in  6  months  from  date,  with  interest  from 
the  date.  On  this  note  are  the  following  en- 
dorsements :  Received,  April  1,  1840,  24  dol- 
lars; Aug.  1,  1840,  4  dollars  ;  Dec.  1,  1840,  6 
dollars;  Feb.  1,  1841,  60  dollars  ;  July],  1841, 
40  dollars;  June  1,  1844,300  dollars  ;  Sept.  1, 
1844,  12  dollars  ;  Jan.  1,  1845,  15  dollars  ;  Oct. 
1,  1845,  50  dollars;  and  the  judgment  is  to  be 
entered  Dec.  1,  1850. 


44  RAINEY'S  IMPROVED  ABACUS. 


CALCULATION. 

The  principal  sum  carrying  interest  from  January  1, 

1840,  $1000  00 
Interest  to  April  1,  1840,  3  months,  15  00 

Amount,         1015  00 
Paid  April  1,  1840,  a  sum  exceeding  the  interest,  24  00 

Remainder  for  a  new  principal,  991  00 

Interest  on- §991   from  April   1,    1840,  to  February  1, 

1841,  10  months,  49  55 

Amount,        1040  55 

Paid  August  1,  1840,  a  sum  less  than  the  in- 
terest due,  $4  00 
Paid  December  1,  1840,   do.              do.  6  00 
Paid  February  1,  1841,     do.  greater  do.               60  00 

70  00 

Remainder  for  a  new  principal,  970  55 
Interest  on  §970  55  from  February  1,  1841,  to  July  1, 

1841,  5  months,  24  26 

Amount,  ~  994  81 

Paid  July  1,  1841,  a  sum  exceeding  the  interest,  40  00 

Remainder  for  a  new  principal,  954  8 1 
Interest  on  §954  81  from  July    1,  1841,  to  June  1, 

1844,  2  years  11  months,  167  00 

Amount,  1121  81 

Paid  June  1,  1844,  a  sum  exceeding  the  interest,  300  00 

Remainder  for  new  principal,  821  81 
Interest  on  $821  81  from  June  1,  1844,  to  October   1, 

1845,  1  year  and  4  months,  65  75 

Amount,      "  887  56 

Paid  September  1, 1844,  a  sum  less  than  the  in- 
terest due,  $12  00 

Paid  January  1,  1845,  do.      do.  1500 

Paid  October  1,  1845,  do.  greater,  with  the  two- 
last  payments,  than  the  interest  then  due,  50  00 

77  00 

Remainder  for  new  principal,  810  56 

Interest  on  $810  56,  from  October  1,  1845,  to   Decem- 
ber 1,  1850,  the  time  when  judgment  is  to  be  en- 
tered, 5  years  and  2  months,  251  30 

Judgment  rendered  for  the  amount,  $1061  86 


PARTIAL    PAYMENTS.  45 

The  following  is  the  rule  of  the  Supreme  Court  of  the 
United  States,  as  given  by  Chancellor  Kent,  Johnson's  Chan- 
cery Reports,  vol.  1st,  page  17;  and  is  adopted  by  most  of  the 
States  of  the  Union,  among  which  are  Massachusetts  and 
New  York: 

SUPREME    COURT    RULE. 

I.  "  The  rule  for  casting  interest,  when  partial  pay- 
ments have  been  made,  is  to  apply  the  payment,  in  the 
first  place,  to  the  discharge  of  the  interest  then  due. 

II.  "  If  the  payment  exceeds  the  interest,  the  surplus 
goes  toward  discharging  the  principal ;  and  the  subse- 
quent interest  is  to  be  computed  on  the  balance  of  the 
principal  remaining  due. 

III.  "  If  the  payment  be  less  than  the  interest,  the 
surplus  of  interest  must  not  be  taken  to  augment  the 
principal ;  but  interest  continues  en  the  former  principal 
until  the  period  when  the  payments,  taken  together,  exceed 
the  interest  due,  and  then  the  surplus  is  to  be  applied 
toward  discharging  the  principal ;  and  interest  is  to  be 
computed  on  the  balance  as  aforesaid" 

A.  gave  to  B.  his  note  for  12,000  dollars;  at  the  expiration 
of  three  months  he  paid  2,000  dollars ;  in  three  months  more 
6,000,  and  at  the  expiration  of  three  months  more,  3,000  dol- 
lars: what  did  he  pay  to  B.  when  the  note  was  taken  up  at 
the  close  of  the  year,  the  note  being  made  on  the  1st  day  of 
January  ?  We  reckon  these  payments  by  the  Supreme  Court 
Rule. 

Principal $12,000 

Interest  on  the  whole  for  three  months  -  -  -       ,180 
Amount  of  principal  and  interest  -----    12,180 

First  payment  to  be  deducted  -------      2,000 

Balance  due  after  first  payment  ------    10,180 

Interest  from  1st  to  2d  payment,  3  months  -        ,152.70 


Amount  to  be  reduced  by  2d  payment  -  -  -    10,332.70 
Second  payment  to  be  deducted  ------      6,000. 


Balance  due  after  2d  payment 4,332.70 

Interest  from  2d  to  3d  payment,  3  months  -  64.99.05 

Amount  to  be  reduced  by  3d  payment  -  -  -  4,397.69.05 

Third  payment  to  be  deducted 3,000. 

Balance  due  after  3d  payment  -  ------  1,397.69.05 

Int.  from  3d  pay't  till  settlement,  3  months  _       20.96.53.57$ 

Balance  due  on  settlement 1,418.65.58.57$ 

4 


46  RAINEY'S  IMPROVED  ABACUS. 

The  following  is  called  the  Commercial  Rule,  and  is  adopted 
by  Vermont: 

COMMERCIAL    OR    VERMONT    RULE. 

Find  the  amount  of  the  whole  debt  until  the  time  of 
settlement ;  then  find  the  amount  of  each  payment  from 
the  time  of  payment  until  the  time  of  settlement ;  add 
these,  and  subtract  the  sum  from  the  former  amount : 
the  remainder  will  be  the  sum  due. 

For  all  payments  made  within  one  year,  this  rule  is  iden- 
tical with  that  of  Connecticut,  which  follows  : 

CONNECTICUT    RULE. 

I.  "  Compute  tlie  interest  on  the  principal  to  the  time 
of  the  first  payment ;  if  that  be  one  year  or  more  from 
the  time  the  interest  commenced,  add  it  to  the  principal, 
and  deduct  the  payment  from  the  sum  total.     If  there  be 
after  payments  made,  compute  the  interest  on  the  balance 
due  to  the  next  payment,  and  then  deduct  the  payment 
as  above;    and  in  like  manner  from  one  payment  to 
another,  until  all  the  payments  are  absorbed ;  provided 
the  time  between  one  payment  and  another  be  one  year 
or  more 

II.  "If  any  payments  be  made  before  one  year's 
interest  has  accrued,  then   compute  the  interest  on  the 
principal  sum  due  on,  the  obligation,  for  one  year,  add  it 
to  the  principal,  and  compute  the  interest  on  the  sum, 
paid,  from  the  time  it  was  paid  up  to  the  end  of  the 
year ;  add  it  to  the  sum  paid,  and  deduct  that  sum  from 
tJie  principal  and  interest  added  as  above. 

III.  "  If  a  year  extends  beyond  the  time  of  payment, 
then  find  the  amount  of  the  principal  remaining  unpaid 
up  to  the  time  of  settlement,  likeioise  the  amount  of  the 
indorsements  from  the  time  they  were  paid  to  the  time  of 
settlement,  and  deduct  the  sum  of  these  several  amounts 
from  the  amount  of  the  principal. 

11  If  any  payments  be  made  of  ct  sum  less  than  the 
interest  arisen  at  the  time  of  such  payment,  no  interest  is 
to  be  computed,  but  only  on  the  principal  sum  for  any 
period.'9  KIRBY'S  REPORTS. 


PHILOSOPHY  OF  RATIO.  47 

Ignorance  of  the  correct  method  of  calcula- 
ting interest  on  partial  payments,  is  the  cause 
of  much  litigation.  Hence,  it  behoves  men 
to  remember,  that  interest  should  be  reckoned 
till  the  time  of  the  first  payment,  and  added 
to  the  principal,  and  the  payment  deducted ; 
provided  the  payment  is  greater  than  the  in- 
terest that  has  accrued.  But  if  the  interest  is 
greater  than  the  payment,  the  payment  must 
be  set  aside,  and  the  interest  reckoned  to 
another  payment ;  or  continuously  from  one 
payment  to  another,  till  the  sum  of  the  pay- 
ments shall  exceed  the  sum  of  the  interest 
accrued.  Then  the  several  sums  of  interest 
should  be  added  to  the  principal,  and  the  sum 
of  the  several  payments  deducted.  The  re- 
mainder will  be  a  new  principal  on  which  in- 
terest runs  till  the  next  current  payment,  or 
till  the  debt^is  paid,  or  judgment  rendered. 
This  method  will  stand  in  the  courts,  of  the 
great  majority  of  the  States  in  the  Union. 


RATIO  AND   PROPORTION. 

That  the  principles  of  subsequent  statements 
may  be  understood,  we  introduce  a  few  re- 
marks on  Ratio  and  Proportion.  The  princi- 
ples and  various  ramifications  of  this  subdi- 
vision of  numbers,  could  not  be  well  developed 
and  elucidated,  in  less  space  than  is  occupied 
by  this  entire  volume.  Hence,  we  attempt 
only  its  outline. 

Ratio  is  the  relation  that  exists  between  different 


48  RAINEY'S   IMPROVED  ABACUS. 

quantities  and  numbers  of  similar  things  ;  and  is 
^  always  expressed  by  an  abstract  number.    Ten 
apples  are  twice  as  many  as  five  apples :  hence, 
the  ratio  or  relation  is  2.     We  have  here  two 
different  numbers  of  the  same  object :  but  it 
could  not  be  said  that,  ten  apples  were  twice 
as  many  as  five  biscuits ;  because  there  is  no 
relation  between  one  apple  and  one  biscuit : 
neither  can  one  be  said  to  be  larger  than  the 
other.     Again,  15  yards  of  cloth  are  five  times 
as  many  as  3  yards  of  cloth :  here  the  ratio  is  5. 
Twenty  gallons  of  melasses   are   4  times   as 
many  as  5  gallons  :  but  20  gallons  of  melasses 
are  not  4  times  as  many  as  5  gallons  of  rum. 
Six  bushels  \vheat  are   one  fifth  times  as 
many   as    30  bushels  :  that  is,  the  ratio  is   \. 
Hence,  the  fact  assumed  in  the  outset  is  proven, 
that  Ratio  is  the  relation  between  similar  things. 
__J^^xatios_make  a  proportion  ;  that  is,  as 
many  times  greater  or  less  as  &  second  thing  is 
than  the  first,  so  many  times  greater  or  less  is  a 
fourth  than  the    third.     These  relations    are 
ascertained,  by  first  ascertaining  the  ratio  that 
exists  between  the  two  numbers  or  quantities 
in  the  proposition,  which   are  alike.     For  in- 
stance :   If  4    yards    of    cloth   cost  $5,  what 
will  12  yards  of  cloth  cost?     Here  the  ratio 
between  the  4  and  12   yards   must  be  obtained 
first.     This  is   done  by  using  a  vertical  line, 
•which  is  considered  the  fulcrum  of  a  scale  or 
balance,  thus, 

The  12  yards  are  placed  on  the  right, 

and   the   4   yards  opposite,  on   the   left. 

Hence,  as  in  the  two  ends  of  the  scale, 

we  compare  their  value.     Four  is  contained 


SIMPLE   PROPORTION   DIRECT.  49 

in  12  three  times :  hence,  the  ratio  is  3.  Now, 
the  four  yards  on  the  left,  are  equal  in  value 
to  5  dollars ;  which  to  balance  the  4  yards, 
must  be  placed,  likewise,  last  on  the  right. 
The  ratio  being  an  abstract  number,  and  being 
on  the  right  with  the  dollars,  may  be  multiplied 
into  the  dollars :  hence,  3  times  $5  are  15 
dollars ;  not  15  bushels,  or  15  of  any  thing  else, 
except  the  denomination  into  which  the  ab- 
stract ratio  is  multiplied. 

From  this,  the  philosophy  of  the  statement 
seems  to  require,  that  the  price  or  denomina- 
tion of  the  answer,  be  placed  last,  on  the  right, 
where  it  can  be  multiplied  by  the  ratio  between 
the  two  quantities  above,  and  increased  or 
decreased  accordingly,  as  the  ratio  is  an  integer 
or  fraction.  From  this,  tao^tojce^j^ju^  the_ 
theory  of  equIIi5ghH^ 

extent,  equals  fKe~price^or  nameTof  answer, 
must  be  placed  opposite,  or  on  the  left :  then  the 
number  which  is  to  be  compared  with  this 
supposed  quantity  on  the  left,  must  be  placed 
on  the  right,  directly  opposite  the  term  on  the 
left,  for  the  purpose  of  ascertaining  the  ratio. 
When  stated  thus,  it  is  not  essential  that  the 
ratio  be  actually  obtained,  by  dividing  one  upper 
term  into  the  other;  for  the  position  indicates 
the  ratio,  and  the  numbers  may  be  canceled  by 
using  one  of  the  numbers  with  either  of  the 
other  two. 

In  applying  this  statement  to  questions  in 
Profit  and  Loss,  the  conditions  of  the  question 
must  be  strictly  noticed :  cost  price,  compared 
with  cost  price;  par  per  cent.,  with  par  per 
cent. ;  reduced  with  reduced,  and  advanced 
3 


50  RAINEY'S  IMPROVED  ABACUS. 

with  advanced  per  centum.  By  this  means, 
and  the  directions  following,  the  statements 
made  in  Profit  and  Loss,  become  rational  and 
easy. 

In  questions  where  any  of  the  terms  are 
fractional,  or  mixed  numbers,  such  mixed  num- 
bers may  be  reduced  to  improper  fractions, 
and  their  numerators  located  in  such  position 
on  the  line,  as  otherwise  the  whole  number 
would  occupy ;  with  all  respective  denomina- 
tors opposite. 

If  7i  yards  cloth  are  worth  $20,  what  will 
18J  yards  come  to?  The  term  of  demand,  or 
which  is  connected  with  the  demand,  is  18-J 
yards.  This  must  be  placed  on  the  right  of  the 
line  first,  in  the  form  of  \5 .  The  numerator 
75  will  consequently  occupy  the  right,  thus, 

The   7i  belongs  to    the   left,  and 
being  made  y ,  we  place  the  nume-* 


£0—5 


9  1 50 


rator  15  on  the  left :  now  the  term 
in  which  the  answer  is  required, 
$20,  is  placed  last  on  the  right :  so 


that  being  multiplied  by  the  ratio  between  the 
two  quantities  above,  it  will  be  made  more  or 
less.  The  question  has  now  lost  its  fractional 
form ;  and  being  easily  stated,  can  be  more 
easily  wrought.  Thus,  if  the  terms  were  such 
that  they  could  not  be  canceled,  they  might  be 
multiplied  on  each  side  ;  and  the  right  divided 
by  the  left,  with  very  little  trouble. 

If  i  of  6  yards  cost  $3,  what  will  i  of  20 
yards  cost  ?  Here,  i  of  20  is  the  demand,  and 
is  placed  on  the  right,  by  its  numerators,  1 
and  20  :  \  of  6  is  the  supposition  of  the  same 


SIMPLE    PROPORTION    DIRECT.  51 

name,  and  is  placed  opposite,  by  its  numerators : 
the  price,  $3,  is  placed  last  on  the  right;  thus, 

This  may  be  proven  by  saying, 
if  i  of  20  cost  7i,  what  will  i  of  6 
cost?  J  of  6  is  the  demand;  i  of 

20  the  same  name,  and  $7i  the     2 $ 

term  of  answer,  all  of  which  are 
located  accordingly,  by  their  nu- 
merators; thus, 


3 
1 

20 


The  result  is  three  dollars,  and  the 
question  is  proven. 


This  department  of  numbers  is  called  "  the 
rule  of  three,51  because  three  terms  are  given 
and  used  in  a  statement  to  find  a  fourth. 
When  this  fourth  term  is  ascertained,  the  pro- 
position is  complete ;  so,  also,  is  the  proportion. 

The  rule  of  three  having  three  terms  given 
to  find  a  fourth,  we  conclude  that,  two  of  these 
being  either  means  or  extremes,  we  may  divide 
either  couplet  by  the  remaining  mean  or  ex- 
treme, and  find  such  mean  or  extreme  as  con- 
stitutes the  fourth  term. 

It  is  a  doctrine  in  proportion,  that  the  product 
of  the  two  means  equals  the  product  of  the  two  ex- 
tremes.  If  this  is  true,  it  is  evident  that  the 
product  of  either  the  means  or  extremes,  is 
the  result  of  the  multiplication  of  the  two 
terms  together  as  factors. 

Suppose  the  extremes  be  multiplied  and  the 
product  divided  by  one  of  the  means;  it  is  evi- 
dent that  the  other  mean  will  be  obtained,  and 


52  RAINEY'S  IMPROVED  ABACUS. 

that  if  it  be  multiplied  with  the  first  mean,  their 
product  will  equal  that  of  the  extremes.  Or, 
if  the  product  of  the  means  be  divided  by  one 
of  the  extremes,  the  other  extreme  will  be  ob- 
tained, which  multiplied  with  the  given  ex- 
treme will  yield  a  product  equal  to  that  of  the 
means. 

In  the  proportion  2  :  4  : :  8  :  16  (two  to  four 
as  eight  to  sixteen),  we  multiply  8  by  4,  mak- 
ing 32;  this  divided  by  2,  one  of  the  extremes, 
gives  16,  the  other  extreme:  or,  divided  by  16r 
gives  the  extreme  2.  If,  again,  the  two  ex- 
tremes, 2  and  16,  be  multiplied,  and  the  pro- 
duct, 32,  divided  by  4,  one  of  the  means,  the 
other  mean,  8,  is  obtained :  or,  if  this  product 
be  divided  by  8,  the  other  mean,  4,  is  obtained. 
This  law  will  be  found  necessary  in  the  expla- 
nation of  Compound  Proportion,  by  cause  and 
effect. 

It  will  require  care  and  attention  in  the 
learner  to  distinguish  the  demand  at  times, 
the  language  of  questions  being  frequently  so 
transposed  as  to  present  seemingly  new  issues 
to  those  not  acquainted  with,  or  accustomed 
to,  analyzing  questions  before  stating  them. 

The  general  nature  of  every  question  must 
be  understood  before  much  can  be  done  by 
any  process  of  solution.  And  this  is  the  only 
service  that  analysis  can  render,  the  elucidation 
of  the  general  bearings  of  the  problem. 

We  have  always  been  opposed  to  the  me- 
chanical routine  by  which  persons  have  gone 
on  from  one  question  to  another,  and  wrought 
some,  merely  because  others  were  wrought  so. 
So  long  as  this  method  of  solving  questions 


SIMPLE    PROPORTION    DIRECT.  53 

by  a  rule,  whose  principles  may  at  first  have 
been  understood,  is  practiced  among  those 
seeking  a  first  knowledge  of  the  science,  we 
may  not  look  for  such  a  knowledge  of  the  sub- 
ject as  will  enable  them  to  apply  the  princi- 
ples to  the  great  every-day  transactions  of 
life. 

We  shall  now  solve  a  variety  of  problems 
and  prove  them ;  that  after  the  path  is  pointed 
out,  the  pupil  may  pursue  it  with  some  degree 
of  pleasure. 

If  10  yards  of  cloth  .cost  $16,  what  will  15 
yards  cost? 

The  demand,  15  yards,  is  placed 
on  the  right,  and  the  term  of  the 
same  name,  10  yards,  opposite; 
while  the  price,  16  dollars,  is 


24 


placed  last  on  the  right,  where  it  can  through 
multiplication,  be  increased  by  the  ratio  of  the 
two  numbers  of  yards.  We  use  the  factor  5 
into  10  and  15.  The  answer  is  24  dollars. 

We  prove  this  by  saying,  if  15  yards  cost 
24  dollars,  what  will  10  cost? 

Thus    it    is    proven    that    10 
yards  cost  the  16  dollars  assumed 

in  the  outset.  ft  1A 

jfjUO 

We  may  now  state  yet  differently,  and  say, 
if  $16  buy  10  yards,  how  many  yards  will 
$24  buy?  thus, 

Here  24  dollars  become  the 
demand,  and  16  the  same  name; 
while  10  yards  is  the  denomina- 
tion of  answer,  and  is  conse- 


15  yds 


quently  placed  on  the  right.     The  ratio  is  ob- 


54  RAINEY'S  IMPROVED  ABACUS. 

tained  between  the  dollars,  and  multiplied  into 
the  10  yards,  increasing  them  to  15  yards. 

Again:    if  $24  buy  15  yards, 

how  many  will  16  buy? 
10yds. 

The  first  form  of  the  question  above  would 
be  stated  and  solved  thus  by  the  old  method: 
10  :  15  : :  16  :  x.  The  second  and  third 
terms,  15  and  16,  would  be  multiplied  togethei 
and  their  product  divided  by  the  first;  thus, 


10  :  15  ::  16 
15 

80 
16 


10)240 


24  dolls. 


By  this  mode  of  pro- 
ceeding, the  figures  are 
accumulated  to  a  large 
number  by  first  multiply- 
ing up,  and  then  dividing 
down  again;  whereas  both 
operations  might  be  dis- 
pensed with,  and  the  term 
of  answer  multiplied  simply  by  the  ratio  be- 
tween the  10  and  15. 

It  appears  singular  that  any  intelligent  au- 
thor should  teach  that  it  was  necessary  in  such 
a  question  as  the  one  just  wrought,  to  mul- 
tiply the  16  dollars  by  the  15  yards,  which  is 
within  itself  impossible ;  for  such  an  operation 
has  no  meaning.  Fifteen  yards  times  16  dol- 
lars will  make  neither  240  yards  nor  240  dol- 
lars. It  may  be  urged  that  it  is  multiplying 
the  dollars  by  the  abstract  terms  of  a  fraction, 
which  represents  the  ratio.  This  may  be  ad- 
mitted in  the  case  of  those  prime  numbers 
whose  ratio  cannot  be  reduced  to  a  single  ex- 
pression. Even  then  we  would  prefer  express- 
ing it  by  a  positive  fraction,  in  a  fraction's 


SIMPLE    PROPORTION    DIRECT.  55 

form,  or  by  a  mixed  number,  either  of  which 
would  indicate  how  many  times  the  dollars,  or 
other  denomination  of  answer,  was  to  be 
taken.  For  when  the  other  process  is  pursued, 
the  whole  idea  of  ratio  between  the  number 
multiplied  and  divided  by,  is  lost;  and  the 
pupil  multiplies  and  divides  just  because  the 
rule  so  teaches  him.  But  tell  him  to  get  first 
the  ratio  between  two  numbers  or  quanti- 
ties, and  then  to  multiply  the  denomination  in 
which  the  answer  is  required,  by  it,  thus  in- 
creasing or  decreasing  it,  and  he  understands 
what  he  is  doing.  He  knows  that  if  the  ratio 
be  larger  than  1,  he  will  have  an  answer  as 
much  larger  than  the  last  term  as  this  ratio 
is  greater;  and  if  the  ratio  be  smaller  than 
unity,  his  answer  will  be  smaller  than  the 
term  to  be  multiplied  by  such  ratio.  But  that 
the  second  and  third  shall  be  multiplied  to- 
gether to  a  product,  but  to  be  divided  down 
again  by  the  first,  is  an  absurdity  that  in  its 
very  form  excludes  the  idea  that  we  should 
keep  constantly  before  us,  that  of  ratio.  It  is 
not  that  the  boy  could  not  understand  ratio, 
but  that  the  mode  of  applying  it  had  so  little 
affinity  to  the  principle,  that  when  the  appli- 
cation was  commenced  in  multiplying  and  di- 
viding, he  found  himself  in  a  new  field  with- 
out any  discern  able  directions  to  his  point  of 
destination.  Such  absurd  regulations  give  the 
young  an  idea  that  nothing  can  or  must  be  at- 
tempted beyond  the  comprehension  of  mere 
mechanical  landmarks;  while  if  he  were  told 
that  he  must  get  the  ratio  between  the  two 
numbers,  and  increase  or  decrease  another  by 


56 


RAINEY'S  IMPROVED  ABACUS. 


it,  the  mist  would  flow  from  his  eyes,  and  de- 
velop the  sunlight  of  unclouded  truth. 

If  5  sheets  of  paper  make  150  pages  of  a 
book,  how  many  sheets  are  required  to  make 
800  pages? 
'  i  800 

It  requires  26|  sheets. 

26f 

If  26f  sheets  make  800  pages,  how  many 
will  5  sheets  make  ? 

In  this  instance  the  demand 
is  5  sheets,  and  the  same  name 
26|  sheets ;  the  latter  is  placed  on 
the  left  after  being  reduced  to 
150  pages.  8_o.  That  is,  the  numerator  is 
placed  on  the  left,  where  the  term  of  same 
name  should  go,  and  3,  the  demominator  on 
the  right.  Knowing  that  26|  sheets  make  800 
copies,  and  that  in  the  use  of  this  800  we  must 
find  how  many  five  sheets  will  make,  we  place 
the  800  last  on  the  right,  that  the  answer  may 
be  in  pages. 

If  240  feet  of  lumber  cost  9  dollars,  how 
many  feet  can  be  purchased  at  18f  dollars? 

A  fp 25        1       The  two  terms  to  be  com- 

~^ 2    pared  are  dollars:  hence  18f 

dollars  is  the  demand.     The 
answer  is  the  number  of  feet 


!500 


which  this  sum  buys,  500  feet. 

If  a  chicken  that  cost  5  cents   is   sold  for  8 
cents,  what  is  the  gain  per  100  cents? 


100—2 


We  know  that  5  cents  gains  3 
cents,  and  inquire,  what  will  100 
cents  or  per  centum  gain?  The  an- 
swer is  60  per  cent. 


SIMPLE    PROPORTION    DIRECT.  57 

If  a  perpendicular  wall  80  feet  high  cast  a 
shadow  at  noon  60  feet  wide,  how  wide  a 
shadow  will  a  perpendicular  church  steeple 
cast,  which  is  240  feet  high  ? 

Here  the  terms  to  be  compared 
are  not  the  wall  and  the  steeple, 
wrhich  are  both  perpendicular,  but 
the  separate  hights  of  these  two 


60 


180 


dissimilar  objects.  The  answer  is  desired  in 
width  or  extent;  this  is  consequently  placed 
last  on  the  right,  where  it  can  be  multiplied  by 
the  ratio  between  the  different  hights.  An- 
swer, 180  feet  wide. 

If  a  shadow  180  feet  wide  be  cast  by  a 
steeple  240  feet  high,  how  high  must  the 
steeple  or  other  perpendicular  object  be  that 
will  cast  a  shadow  60  feet  wide? 

Here  the  widths   are   com-    $ — ^01^0 
pared,  and  the  answer  is   ob-  ^0 — 8 

tained  in  hight.  gO 

How  many  gallons  of  oxygen  will  be  neces- 
sary to  make  720  gallons  of  water,  if  9  gal- 
lons of  water  require  8  gallons  of  oxygen  ? 

We  compare  water  with  water. 

The  answer  is  640  gallons  oxygen. 

b40 

If  nine  gallons  of  water  require  1  gallon  of 
hydrogen,  how  much  hydrogen  is  required  to 
make  720  gallons  of  water? 

0  ,W— 8 
The    answer    is    80     gallons    of        \ 

hydrogen.  — 

If  after  I  see  the  flash  of  a  cannon,  I  hear- 


58  RAJNBY'S  IMPROVED  ABACUS. 

the  report  in  4  minutes,  how  far  will  it  be  off, 
if  sound  flies  at  the  rate  of  1142  feet  per 
second  ? 


11240 
3  1142 


1760 


51TT 


Four  minutes  make  240  seconds, 
which  is  the  demand,  while  1  se- 
cond is  the  same  name,  and  1142 
feet,  the  distance  which  sound  flies 
in  the  one  second,  is  the  term  of 


19 


16 


171 


answer.  We  say,  how  many 
yards  will  these  feet  make,  if  3  feet  opposite 
make  1  yard  :  and,  again  how  many  miles 
will  these  yards  make  if  1760  yards  opposite 
make  one  mile?  Thus  three  distinct  propor- 
tions are  combined  in  one,  which  produces  no 
inconsiderable  economy  in  the  use  of  figures. 
If  |  of  a  pound  of  butter  costs  4j  cents, 
what  will  li  pounds  cost? 

The  demand  is  placed  on  the 
right  as  |,  the  same  name  on  the 
left  as  f,  and  the  price  last  on  right 
as  y.  It  is  seen  that  the  numera- 
tor of  the  same  name  occupies  the 
left,  and  its  denominator  the  right. 
None  of  the  numbers  in  this  question  can  be 
canceled;  yet  no  sane  man  will  dispute  that 
the  question  is  stated  in  much  better  form  than 
by  the  old  rules;  for  here  the  pupil  sees  at  a 
glance  what  terms  must  be  multiplied  togeth- 
er, and  what  divided  by,  from  their  very  loca- 
tion on  the  two  sides  of  the  line ;  which  is  not 
the  case  in  the  old  method.  Then,  the  state- 
ment being  rational  and  easy,  it  is  far  prefer- 
able, though  not  a  figure  can  be  canceled. 

If  4j  pounds  of  wool  cost  30  cent/,  what 
will  18|  pounds  come  to? 


SIMPLE    PROPORTION    DIRECT.  59 

This  example  may  easily  be 
proven  by  using  the  answer,  in 
making  inquiry  with  regard  to 
some  other  portion  of  the  ques- 
tion. 

"A  hare  starts  12  rods  before  a  hunter,  and 
ecuds  away  at  the  rate  of  10  miles  an  hour: 
now,  if  the  hunter  does  not  change  his  place, 
how  far  will  the  hare  get  before  him  in  45 
seconds?" 

The  demand  is  here  45  seconds,  which  we 
place  on  the  right :  now,  that  we  may  reduce 
the  hours  to  seconds  and  use  them  on  the  left, 
we  place  opposite  the  45,  60  seconds,  which 
make  a  minute ;  while  the  one  minute  is 
placed  on  the  right  last,  as  the  denomination 
of  answer.  But  that  we  may  continue  on, 
and  reduce  these  minutes  to  hours,  we  place 
opposite  this  one  minute  60  minutes,  which 
make  an  hour,  and  which  is  equivalent  to  10 
miles  running  of  the  hare.  As  this  60  minutes 
equals  the  10  miles  running,  we  must  place  the 
latter  last  on  the  right;  for  we  wish  the  an- 
swer in  distance.  Now,  it  is  evident  that  if 
the  question  were  wrought  without  going  far- 
ther, the  answer  would  be  in  miles;  but  wish- 
ing it  in  rods,  we  place  1  mile  opposite  10,  and 
8  furlongs  on  the  right,  saying,  how  many  fur- 
longs will  all  the  miles  on  the  right  make,  if 
1  mile  makes  8  furlongs  ?  We  say  again, 
how  many  rods  will  all  of  these  furlongs 
make,  if  1  furlong  opposite  the  8,  make  40 
rods ;  which  being  the  term  of  answer,  is 
placed  last  on  the  right;  thus, 


60 


RAINEY'S  IMPROVED  ABACUS. 


40 
12 


152  rods. 


The  12  rods  which  the 
hare  had  in  the  start,  added 
to  this,  makes  the  answer,  52 
rods. 


Again:  "If  a  dog  by  running  16  miles  an 
hour  gain  on  a  hare  6  miles  every  hour,  how 
long  will  it  take  him  to  overtake  her,  if  she 
has  52  rods  the  start?" 

The  demand  is  52  rods,  and  40  rods 
opposite  equal  1  furlong,  8  furlongs 
equal  1  mile,  6  miles  equal  60  minutes 
of  running,  and  1  minute  equals  60 
seconds :  hence  the  answer  is  97i 
seconds. 

Again:  "A  hare  starts  12  rods  before  a 
greyhound,  but  is  not  perceived  by  him  until 
she  has  been  up  45  seconds;  she  scuds  awray 
at  the  rate  of  10  miles  an  hour,  and  the  dog 
after  her  at  the  rate  of  16  miles  an  hour: 
what  space  will  the  dog  run  before  he  over- 
takes her?" 


40t52 

6|i 

160 

60 


60 
60 


195 

1 

16 


40 


What  will  97i  or  'f5  seconds  be,  if 
60  seconds  make  1  minute,  and  60 
minutes  equal  16  miles  running  of 
the  dog,  and  1  mile  equal  8  furlongs, 
and  1  furlong  equal  40  rods,  the  de- 
nomination of  the  answer?  The  dog 
will  run  1381  rods 

These  questions  cannot  be  fully  elucidated 


SIMPLE    PROPORTION    IN    FRACTIONS. 


61 


in  this  little  work,  and  are  given  merely  to  in- 
dicate the  capabilities  of  this  beautiful  system 
of  statement  and  solution. 


If  £  of  a 
5 


pound  of  rice  feed  3   men,  how 


many  will  ~  pounds  feed? 

We  treat  complex  fractions  in 
the  statement  as  all  others,  plac- 
ing the  demand  on  the  right,  same 
name  on  the  left,  etc.  In  the  de- 
mand, the  numerator  of  the  nume- 
rator is  placed  on  the  right,  and 
the  numerator  of  the  denominator 
on  the  left,  with  all  respective  de- 
nominators opposite  their  numerators. 

of    2   yards   of  muslin 


2A 


30 


If 

11  15 


4f 


be   worth   •— 


3 


yards  of  gold  lace,  how  many  yards  of  gold 

40 
lace  will  pay  for  —  of  12  yards  of  muslin  ? 

j         £40—2 

4 

Five  times  3  equal  15;  3  into 
9  three  times,  and  3  times  2  equal 
6;  5  into  10  twice,  and  into  25 
five  times.  4X4X2X2X12=768, 
which  divided  by  5  gives  153f 
yards  of  gold  lace  for  the  answer. 


5768 


153J 

All  such  propositions  as  these,  though  not 
practical  in  their  bearing,  will,  nevertheless, 
5 


62  RAINEY'S  IMPROVED  ABACUS. 

afford  interesting  entertainment  to  those  study- 
ing for  the  mere  beauty  of  theory,  while  it  is 
believed  that  a  sufficient  number  of  examples 
have  been  given  to  meet  all  practical  purposes. 
We  are  aware  that  many  authors  consider 
the  treatise  of  Proportion  under  two  heads,  as 
superfluous.  It  may  be  superfluous  when 
treated  mechanically  ;  but  when  cause  and  ef- 
fect, as  the  bases  of  these  principles,  are  de- 
veloped, they  induce  the  division  on  natu- 
ral, rational,  and  necessary  grounds. 

SUMMARY    OF    DIRECTIONS    FOR    DIRECT    PROPORTION. 

Ascertain  first  the  term  of  Demand: 
Place  the  Demand  Jtrst  on  the  right  : 
Place  the  term  of  the  Same  Name  opposite  the 
Demand,  on  the  left: 

Place  the  term  in  which  the  answer  is  required, 
last  on  the  right,  and  the  answer  will  be  in  the 
same  denomination. 

In  all  fractional  terms,  the  Numerator  must  oc- 
cupy the  side  of  the  line  ordinarily  assigned  to  the 
integer. 


PROFIT  AND  LOSS. 

Profit  and  Loss,  as  constituting  a  depart- 
ment of  the  Arithmetic,  relate  to  the  various 
gains  and  losses  of  general  business  transac- 
tions. They  are  reckoned  in  two  ways:  spe~ 


THEORY  OF  PROFIT  AND  LOSS.  63 

cificcdly,  and  at  a  certain  per  centum.  A  specific 
gain,  is  where  any  gross  quantity,  at  a  given 
cost  price,  is  sold  at  some  other  price,  without 
reference  to  any  regular  sum  of  profit :  as  30 
galls,  of  wine,  which  cost  $19,  being  sold  at 
$31,40;  hence,  the  gain  is  $12,40,  on  the  cost. 

Gain  or  loss  per  cent.,  is  where  a  certain 
sum  is  gained  on  the  hundred ;  for  instance,  a 
bill  of  goods  costs  $50 ;  but  it  is  desired  to  sell 
it  at  30  per  cent,  profit;  consequently,  it  must 
be  sold  at  $65 ;  the  15  dollars  gained,  being  a 
systematic  profit. 

Some  business  men  adopt  a  conscientious 
mode  of  trading,  and  are  willing  to  gain  such 
a  per  cent,  profit,  as  is  just  and  reasonable; 
while  others,  of  less  principle,  and  more  exor- 
bitant in  their  exactions,  make  the  ignorance, 
credulity,  or  necessity  of  the  purchaser,  their 
only  standard;  and  grasp  any  advantage  that 
circumstances  may  afford  them. 

Some  individuals  calculate  their  profits  and 
losses  with  reference  to  time;  while  others, 
and  indeed  the  great  mass  of  business  men, 
regard  only  the  simple  transaction.  That  we 
should  reckon  with  regard  to  the  length  of 
time  occurring  between  the  transactions  of 
buying  and  selling,  appears  very  reasona- 
ble, when  we  consider  that,  all  capital,  as  a 
medium  of  gain,  has  delegated  to  it,  all  the 
productive  capacities  of  active  individual  effort. 

The  general  and  unavoidable  expenses  of 
every  establishment,  require  the  influx  of  a 
constant  gain  to  preserve  the  capital  stock 
entire.  Hence,  the  capital  stock  that  a  man 
invests  in  business,  must  yield  periodically,  and 


64  RAINEY'S   IMPROVED  ABACUS. 

at  short  intervals,  a  sufficient  amount  of  profit 
to  sustain  these.  And  if  the  capital,  which 
may  be  the  only  gain  producing  element  in  a 
man's  business,  be  permitted  to  lie  during  a 
long  interval,  between  the  purchase  and  sale, 
the  profit  on  the  sale  when  made,  must  be 
larger  in  a  degree  corresponding  to  the  length 
of  time  thus  invested ;  or  the  deficit  in  meeting 
the  expenses,  must  convert  this  seeming  gain 
into  a  positive  loss. 

Let  us  suppose  money  worth  10  per  cent,  at 
interest.  The  business  man  who  has  2000 
dollars  may  easily  lend  it,  and  realize  at  the 
end  of  the  year,  200  dollars  profit.  But  he 
prefers  investing  it  in  goods,  which  he  will  sell 
at  10  per  cent,  profit.  If  the  stock  of  goods 
be  sold  in  6  months,  he  may  invest  again,  and 
sell  again,  in  6  months.  Thus  he  would  make 
400  dollars  clear  money.  But  suppose  he  sell 
his  first  stock  at  the  end  of  the  year,  at  the  10 
per  cent,  profit;  he  will  gain  200  dollars.  Sup- 
pose, again,  he  sells  only  half,  it  is  manifest 
that  he  gains  only  100  dollars;  or  that,  if  he 
sell  the  whole  in  a  year  and  a  half,  his  gain 
will  be  200  dollars  in  H  years,  or  6f  percent, 
per  annum. 

Profit  and  Loss  may  be  divided  into  five 
distinct  varieties.  They  are, 

First:  To  find  how  an  article  must  be  sold 
to  gain  or  lose  a  certain  per  centum;  or  to  find 
the  sum  of  gain  or  loss,  at  a  specified  per 
centum. 

Second:  To  find  the  rate  per  cent.,  profit  or 
loss,  when  an  article  is  purchased  at  one  price 
and  sold  at  another. 


VARIETIES  OF  PROFIT  AND  LOSS.  65 

Third:  To  find  the  cost  price,  when  an 
article  has  been  sold  at  a  certain  per  cent., 
gain  or  loss. 

Fourth:  To  find  the  rate  per  cent.,  gain  or 
loss,  when  an  article,  sold  at  a  certain  price, 
with  a  specified  gain  or  loss,  is  advanced  or 
reduced  to  yet  another  price. 

Fifth :  To  find  the  selling  price  of  an  article, 
whose  cost  price  is  affected  by  commission, 
premium,  discount,  loss  or  drawback,  gain  or 
loss  per  cent.,  &c. 

All  operations  coming  under  any  of  these  five 
heads  in  Profit  and  Loss,  depend  primarily  on 
the  principles  of  ratio  and  proportion.  Ope- 
rations in  specific  profit  and  loss,  depend  on 
addition  and  subtraction,  only.  Per  centum 
being  the  great  acknowledged  basis  in  the  rule, 
everything  is  in  a  ratio,  greater  or  less  than 
100.  The  arbitrary  rules  in  all  of  the  old 
books,  on  this  subject,  have  tended  to  make 
this  simple  and  beautiful  department  of  Arith- 
metic, complex;  and  even,  in  many  cases, 
unintelligible;  whereas,  when  the  relations 
and  bearings  of  the  proportional  principles 
involved,  are  demonstrated,  we  see  a  harmony 
and  system,  a  regularity  and  order,  that  no 
other  portion  of  the  science  can  excel.  Calcu- 
lations in  profit  and  loss  are,  however,  of  a 
more  apparently  abstract  nature,  that  in  the 
general  calculations  of  ordinary  business  con- 
cerns. Hence,  it  becomes  necessary,  that  we 
make  nicer  distinctions  between  the  terms,  and 
their  specific  names  and  qualities,  than  where 
material  objects,  such  as  yards,  Ibs.,  &c.,  are 
concerned.  Here,  we  have  to  compare  cost 


66  RAINEY'S  IMPROVED  ABACUS. 

price  with  cost  price,  selling  price  with  selling 
price,  advanced  with  advanced,  and  reduced 
\vith  reduced  price.  And,  too,  these  distinctions 
are  vital  and  necessary;  for  they  constitute 
supposition,  demand,  and  term  of  answer. 
Some  authors  work  profit  and  loss  by  a  system 
of  decimals,  which,  although  correct,  yet  ob- 
scures entirely  the  principles  of  ratio ;  making 
the  statement,  as  well  as  the  work,  depend  on 
a  mechanical  arrangement. 


VARIETY  FIRST. 

If  a  Ib.  of  coffee  cost  10  cents,  for  how  much 
must  it  be  sold  to  gain  20  per  cent.?  We  may 
remark  here,  that  the  cost  price  of  an  article 
is  called  the  "par"  price;  par,  as  meaning  the 
equal  of  something  else,  or  the  equal  of  the 
price.  When,  therefore,  we  say  the  par  of  per 
centum,  we  mean  the  100,  without  increase  or 
decrease,  as  its  own  established  rate  indicates. 
When  we  say  the  par  price  of  an  article,  we 
mean  the  cost.  Thus  par  price,  and  par  per 
centum,  may  be  considered  the  same  thing. 
When  100  is  compared  with  any  specific  thing, 
as  cents,  in  the  above  case,  it  loses  its  abstract, 
and  assumes  the  denominate  form  of  the 
specific  thing  with  which  compared.  Hence, 
in  the  case  above,  we  may  say  that  the  100 
becomes  100  cents •,  because  compared  with  10 
cents.  Now,  the  position  assumed  at  first  was, 
that  per  centum,  or  100,  gained  20,  which, 
changed  from  this  abstract  form  to  the  denom- 
inate, is  100  cents  giving  20  cents,  or  100  cents 
being  advanced  to  120  cents;  for,  if  100  cents, 


TO  FIND  SELLING  PRICE.  67 

or  anything  else,  gain  20,  they  must  be  ad- 
vanced to  120. 

Now,  we  know  that  100  is  the  par,  or  cost, 
or  first  value  of  per  centum ;  but  120  is  the 
advanced  price  or  value  of  it.  Ten,  we  know, 
likewise,  is  the  price  of  the  coffee.  Now,  our 
demand,  10  cents,  cost  price,  is  placed  on  the 
right;  100,  or  per  cent.,  opposite,  on  the  left; 
and  120,  advanced  price,  last  on  the  right :  con- 
sequently, the  answer  would  be  in  the  advanced 
or  selling  price  of  the  coffee.  Therefore,  the 
ratio  is  obtained  between  the  10  cents  and  the 
100  cents;  and  the  selling  price,  120,  is  multi- 
plied by  it.  The  ratio  is  TJ7 ;  consequently,  T^ 
of  120,  or  12,  is  the  selling  price  of  the  coffee, 

thus :  \id\ib  id 

We   may,  instead   of  placing   the          |i20 

advanced  or  selling  price  last  on  the    T~- 

right, place  the  gain  of  100  there;  get 
the  answer,  in  gain,  and  add  it   to  the    cost 
price.    If  100  gain  20,  what  will  10  gain  ?    We 
add  this  gain,  2,  to  the  cost  price,  and  i^xxi^x 
have,  as  before,  12,  selling  price.     The          K^ 

latter  process,  however,  is  no  more  per-    

spicuous,  while  it  is  more  tedious  than  ' 
the  former.  Hence,  it  is  best  to  find  the  sell- 
ing price,  by  placing  the  advanced  or  reduced 
per  cent.,  last  on  the  right.  Let  us  now  find 
how  we  will  sell  this  1  Ib.  to  lose  20  per  cent. 
We  know  that  if  per  cent,  loses  20,  it 
will  be  reduced  to  80;  so,  if  100  cost, 
be  80,  selling  price,  what  will  be  the 
Belling  price  of  10,  cost? 

Let  us  find  the  result,  by  the  2d  process,  as 
4 


68  RAINEY'S  IMPROVED  ABACUS. 

before  ;  by  ascertaining  first,  the  loss,  and  then 
subtracting  it  from  the  10. 

What  will   10  lose,  if  100  lose  20? 
We  have  2  as  before,  which  subtract- 
12       ed,  leaves  8. 

One  thing  may  be  noticed  very  particularly, 
in  the  above ;  that  the  sum  of  profit  or  loss  on 
any  given  sum,  as  the  100  above,  is  the  same.  In 
this  case  10  either  gained,  or  lost  2.  This  leads 
us  to  remark,  that  a  large  sum  will  gain  or  lose 
more  than  a  smaller.  While  10  either  gains 
or  loses  2,  12  will  gain  or  lose  more  than  2  : 
20  cents  will  gain  or  lose  twice  as  much  as 
10  cents,  while  5  cents  will  gain  or  lose  only 
one  half  as  much.  We  make  these  remarks, 
because  there  are  some  who  may  not  appre- 
hend the  difference  between  the  gains  or  losses 
on  larger  or  smaller  sums.  Such  think  that 
if  20  per  cent,  gain  will  advance  10  to  12,  as 
a  matter  of  course,  20  per  cent,  loss  will  reduce 
12  to  the  10.  The  difference  consists  in  this, 
that  10  will  gain  2,  and  advance  to  12:  but 
12,  losing  more  than  2,  will  be  reduced  to  9|. 
It  will  be  well  for  the  pupil  to  bear  this  differ- 
ence in  mind,  as  its  importance  will  be  seen 
in  the  calculations  that  follow.  We  might 
prove  the  foregoing  questions,  by  asking  the 
cost  price,  after  knowing  the  advanced  pricer 
12 :  but  as  this  operation  comes  under  variety 
3d,  we  will  defer  proof  till  we  treat  of  that 
variety.  It  may  be  proven  by  variety  2d,  by 
finding  the  rate  per  cent.  From  the  foregoing 
we  conclude,  that, 

To  find  hoiv  an  article  must  be  sold,  to  gain  or 


PROFIT   AND   LOSS—VARIETY   FIRST. 


69 


10015 
24225 


lose  a  given  per  cent.;  place  the  cost  price  on  the 
right,  for  the  demand:  100  on  the  left,  for  the  same 
name;  and  100,  increased  by  the  gain  per  cent, 
added,  or  diminished  by  the  loss  per  cent,  subtract- 
ed, last  on  the  right,  for  the  term  of  answer. 

Purchased  sugar  at  5  cents  per  lb.:  how  must 
it  be  sold  to  gain  12i  per  cent.? 
Here,  5  cents  is  the  demand;  100  the 
same  name;  and  112i,  the  name  of 
answer.  The  latter  is  reduced,  ma- 
king --§-,  and  is  stated  as  above. 

Again  :  how  must  cloth  that  cost 
$7|  per  yard,  be  sold  to  lose  40  per 
cent.?  that  is,  what  will  $y  be  re- 
duced to,  if  $100  are  reduced  to  60? 

If  a  horse  cost  $50,  how  must  he 
be  sold  to  gain  50  per  cent.  ?  Here, 
the  demand  is  $50,  100  the  same 
name,  and  150,  the  term  of  answer. 


100 


15 
60 


$|4i 


150 


|75 


We  find  that  50  per  cent,  advances  to  $75  : 
or,  that  it  gains  $25. 

Let  us  see  if  selling  another  horse 
that  cost  $75,  at  50  per  cent,  loss, 
will  bring  75  back  to  50.  It  is  per- 
ceived here,  as  in  a  former  question, 


100|75 
150 


1374 


that  50  per  cent,  loss  on  $75,  brings  75  down 
to  $37i  ;  for  while  50,  at  50  per  cent.,  gains 
or  loses  25,  75,  at  50  per  cent.,  gains  or  loses 
37|.  This  difference,  and  the  reason  for  it, 
are  too  palpable  to  need  further  comment. 

If  8  yards  of  cloth  cost  $20,  for  how  much 
must  200  yards  be  sold,  to  gain  20  per  cent.  ? 
Here,  it  is  manifest  that  the  statement  is  made, 
first  by  proportion,  thus, 


70 


RAINEY'S  IMPROVED  ABACUS. 


8'200         This  being  the  cost  of  200  yards, 

1 20       we  £°  on  anc^  say>  wnat  will  these 
$500  be  advanced  to,  if  100,  or  per 
100  50C     cent.?   be    advanced   to    120?     The 
120     answer  is   600.     It  is  wholly  unne- 
$1600     cessary  to  make  two  separate  state- 
ments in   this    case :    for,  knowing 
that  the  answer  is  involved  in  the  first  state- 
ment, that  is,  that  the  number  of  dollars  which 
the  200  yards  would  cost,  could  be  found  by 
working  the  question,  we  say,  what  will  this 
supposed  answer  or  sum,  be  yet  farther  ad- 
vanced to,  as  the  demand,  if  100  opposite,  be 
made  120  ?  thus, 

The  $100,  or  per  cent.,  placed 
on  the  left,  is  placed  opposite  the 
$20,  which  term  represents  the 
answer,  and  which  is  the  demand. 


— 6 


6 

18 

1 

8 

100 


15 


9 

1200 

275 


$1775, 


If  i  of  |  of  a  farm  cost  $1200, 
for  how  much  must  1  of  T|  of  the 
same  be  sold,  to  gain  37i  per 
cent.  ?  The  statement  of  the 
question  made,  by  proportion, 
nothing  more  is  necessary  than 
to  place  100  opposite  dollars,  or 
involved  answer,  and  137i,  or 
~-j  on  the  right. 

I  have  300  Ibs.  bacon,  which  cost  5  cents 
per  pound  :  how  must  I  sell  the  whole,  to  gain 
25  per  cent.  ?  It  is  necessary  in  the  first  place, 
to  find  the  cost  of  the  whole  bacon ;  conse- 
quently, we  say,  if  1  Ib.  cost  5  cents,  what 
will  300  Ibs.  cost  ?  thus, 


COMBINATION  OF  STATEMENT.  71 

Then  we  say,  what  will  all  the  cts.        .,     , , 


5 

125 


paying  for  the  bacon,  be  advanced 
to,  if  100  be  advanced  to  125? 

Bought  200  galls,  oil,  at  60  cents  », «-«-,— 
per  gallon  ;  I  lost  40  galls,  and  wish 
to  know  how.  I  must  sell  the  remainder  to  gain 
30  per  cent,  on  the  whole  investment.  What 
will  200  galls,  cost  if  1  gallon  cost  60  cents? 
Then  we  know  that  after  deducting  40  gallons 
loss,  we  have  but  160,  which  this  sum  of 
money,  on  the  right,  has  paid  for.  Supposing 
this  on  the  right  to  be  the  price  of  the  160 
galls.,  we  say,  what  will  one  gallon  cost  on 
the  right,  if  160  opposite,  cost  these  cents  : 
the  result  would  be  the  advanced  price  at 
which  1  gallon  of  the  160  would  be  sold; 
so  that  the  whole  sale  would  bring  the  money 
first  invested  in  the  200.  We  then  say,  what 
will  this  price  be  yet  farther  advanced  to,  if 
100,  or  per  cent.,  opposite,  be  130?  We  get 
the  answer  in  the  price  of  1  gallon,  at  30  per 
cent,  profit,  thus, 


160 
100 


200 

60 

130 


In  the  case  above,  all  that  is  ne- 
cessary, is  to  suppose  that  the  de- 
mand does  exist  on  the  right ;  for  we  i  ,^ , 
know  that  this  demand  is  but  the  ' 
result  of  another  proportion,  preceding,  which 
we  could  easily  ascertain  by  making  the  sepa- 
rate statement.  The  demand,  1  gallon,  is 
merely  supposed ;  for  it  is  wholly  unnecessary 
to  assign  a  place  to  a  unit,  which,  so  far  as  the 
work  is  concerned,  is  useless. 


72  RAINEY'S  IMPROVED  ABACUS. 


SECOND    VARIETY. 

If  I  buy  a  Ib.  of  butter  for  10  cents,  and  sell 
it  for  15  cents,  what  do  I  gain  per  cent.  ?  The 
demand  here  is,  what  will  per  centum,  or  100 
gain,  if  10  cents  opposite,  gain  5?  The  ques- 
tion is  not  what  will  100  be  advanced  to,  if  10 
be  advanced  to  15;  but  to  know  how  much 
100  will  gain.  Stating  accordingly,  the  answer 
will  be  the  gain  of  100,  or  the  rate  per  cent, 
profit.  It  is  supposed,  not  unfrequ  ently ,  that 
the  per  cent,  must  be  calculated  on  the  selling 
price,  which,  in  the  case  cited,  would  present 
this  absurdity:  if  15  cents  gain  nothing,  what 
will  100  cents,  or  per  cent.,  gain?  We  know 
that  the  gain  has  been  effected  by  the  use  of 
the  10  cents;  and  if  the  10  cents  have  gained 
5  others,  certainly  in  the  same  ratio,  100,  or 
per  cent.,  which  are  10  times  as  many,  will 
gain  10  times  as  much  as  5,  which  is  50  per 
cent.  We  could  .find  this  rate  per  cent,  profit 
in  the  following  way:  if  10  be  advanced  to 
15,  what  will  100  be  advanced  to?  In  this 
case,  the  cost  price,  10,  is  added  to  the  gain,  5, 
to  make  the  advanced  price ;  therefore,  the  par 
per  centum,  100,  will  be  added  to  the  gain  of 
the  100,  to  make  advanced  per  cent.  Hence, 
the  necessity,  if  the  question  be  wrought  thus, 
of  subtracting  100  from  the  answer.  This 
mode  of  finding  per  cent,  is,  however,  neither 
direct  nor  natural.  Here,  it  is  seen,  that  as  we 
would  subtract  the  par,  10,  to  leave  the  gain, 


RATE  PER  CENT.  PROFIT  AND  LOSS 


BO  we  would  subtract  par  per  cent, 
to  leave  the  gain  per  cent.  This  is 
presented  only  for  its  theory,  which 
will  be  found  applicable  in  variety 
fourth.  Fifty  per  cent,  answer. 


in 


inn 


—  — 


- 
I    50 


40  400 
2 


20 


We  now  revert  to  variety  first,  and  knowing 
that  the  one  Ib.  of  coffee,  which  cost  10  cents, 
was  sold  at  12,  to  gain  20  per  cent.,  we  will 
see  if  this  is  correct.  When  purchased  at  10 
and  sold  at  12,  the  gain  was  2  :  now, 
if  10  cents  gain  2,  what  will  100  cents 
gain?  Twenty,  which  is  20  per  cent., 
is  the  answer. 

Suppose  the  question  above  were  thus  :  if 
10  cents  lose  2,  what  is  the  loss  per  cent.?  It 
is  evident  that  it  would  be  precisely  the  same 
operation,  and  that  20  per  cent,  would  be  the 
result.  Again  :  If  I  buy  raisins  at  7|  cents 
per  Ib.,  and  sell  them  at  10  cents,  what  do  I 
gain  per  cent.?  The  first  thing  to  do  in  all 
such  cases  is,  evidently,  to  find  the  gain  or  loss 
on  the  cost,  and  say,  if  the  cost  has  given  this 
gain,  what  will  per  centum  give?  Now,  7-J- 
cents  gain  2i  ;  the  question  is  stated 
accordingly;  disposing  of  fractions 
in  the  usual  manner.  The  result,  33  J 
per  cent.,  is  evidently  correct,  if  we 
consider  that  2i  being  i  of  7i,  the 
answer  should  likewise  be  J  of  100. 

If  a  horse  is  bought  for  40  dollars, 
and  sold  for  80,  what  is  the  gain  per 
cent.  ?  If  40  gain  40,  per  cent,  will 
gain  100.  The  result  is  100  per 
centum. 


15 


100 


100 


RAINCY  S  IMPROVED  ABACUS. 


From  the  foregoing  \ve  conclude  that,  To 
find  the  rate  per  cent.,  profit  or  loss,  when  an 
article  is  purchased  at  one  price  and  sold  at 
another; 

Ascertain  the  gain  or  loss  on  tfie  cost  price,  by 
subtraction:  make  100  the  demand:  the  cost  price, 
the  same  name;  and  the  gain  or  loss,  the  name  of 
answer,  last  on  the  rigM.  The  answer  will  be  the 
gain,  or  loss,  per  cent. 

^  k^l  °*  g00^3  cost  2400  dollars, 
and  was  sold  for  3000;  what  was 
the  gain  per  cent.  ?  The  2400  gain- 
ed 600;  hence,  the  per  cent,  is  %J 


2400  100 
600 


[25 
Bought  cloth  at  4j  dollars,  and  sold  it  at  4f ; 


how  much  per  cent,  was  gained  or  lost  ?  We 
find  that  45  dollars  lose  J  of  a  dollar  :  hence, 
we  inquire,  what  will  1  00  lose  ?  The  rate  of 
loss  is  21-f  per  cent.,  thus, 


19 
8 


100 


100 
19 


19 
1850 


844 


75 


|25 
100 


4 
425 


be  proven  by  Variety  1st, 
by  asking,  to  what  price  will  43  be 
reduced,  to  lose  2i-|  per  cent.  ?  That 

I  is,  what  will  y  be  reduced  to,  if  100 
be  reduced  to  97T7^  or  ^ff-?  Four 

1  and  five-eighths  is  evidently  the 
answer;  for  this  was  the  price  at 
which  it  was  first  sold,  at  a  loss. 

Ribbon  that  cost  6  cents  per  yard, 
is  sold  at  7^  cents,  what  is  the  gain 
per  cent.? 

Cloth  that  cost  183  cents  per  yard, 
is  sold  at  12i  cents:  what  is  the 
loss  per  cent.?  The  183  lose  6i; 
therefore,  100  will  lose  33:,  which 
is  the  rate  per  cent.  loss. 


PROFIT  AND  LOSS :  COST  PRICK.  75 

It  must  be  observed  here,  that  loss,  as  well 
as  gain,  is  made  on  the  cost  price. 

THIRD    VARIETY.* 

Sold  a  yard  of  cloth  for  250  cents,  and 
thereby  gained  50  per  cent.;  what  did  it  cost?. 
It  is  manifest  that  50  per  cent,  was  calculated 
on  the  cost  price,  and  then  added  to  it,  for  the 
selling  price ;  so  that  to  take  50  per  cent,  from 
the  selling  price,  which  is  considerably  larger 
than  the  cost  price,  would  be  taking  off  a 
larger  sum  than  the  cost  price,  at  50  per  cent., 
would  give.  It  is,  therefore,  necessary  to  find 
the  cost,  and  then  50  per  cent,  reckoned  on 
this  and  added,  would  make  the  selling  price. 
The  250  cents  are  the  advanced  price,  from 
•which  we  wish  to  deduct  the  per  centum  that 
has  been  added.  To  do  this,  we  must  compare 
it  with  the  advance  value  of  per  cent.,  or  with 
100,  advanced  by  the  rate,  added  to  it.  Now 
the  rate  50,  when  added  to  100,  makes  150,  for 
the  advance  or  amount  of  per  cent.  If  this 
be  the  advance  or  amount  of  per  cent.,  to  find 
the  cost  or  par,  we  must  reduce  it  to  100.  We 
have,  therefore,  150  advance  per  cent,  to  com- 
pare by  ratio  with  250,  advanced  price  of 
cloth,  and  conclude,  that  if  150  advance,  be 
reduced  to  100,  par  or  cost,  250  advance 
must  be  reduced  in  the  same  ratio,  to  find  its 
par  or  cost.  Hence,  we  make  250  the  demand, 
150  the  same  name,  and  100  the  term  of 
answer:  thus, 


76 


RA1NEY;S   IMPROVED   ABACUS. 


150 


250 
100 


In  reducing  the  150  above,  it  is 
not  the  150  losing  50  per  cent. ;  for 
this  would  reduce  it  to  75,  instead 
$|l,66f  °f  10°  :  but  it  is  150  losing  the  gain, 
being  brought  back  to  the  cost  or 
sum,  that  first  gained  it.  If  100,  at  50  per 
cent.,  be  150,  certainly  100  of  it  is  original 
value,  and  50  gain ;  both  making  the  amount 
150.  So,  likewise,  with  the  cloth  ;  166|  is  the 
first  value,  and  83  J  the  gain:  both  together, 
making  the  amount  or  advanced  price,  250. 
Now,  by  proportion  we  find,  that  if  100  would 
gain  $50,  166|  would  gain  83|.  So,  it  be- 
comes reasonable,  that  in  reducing  an  article 
that  has  been  sold  at  an  advanced,  to  its  cost 
price,  to  compare  advanced  price  of  the  article 
with  advanced  per  cent.  We  may  now  prove 
this  by  both  preceding  Varieties. 


100 


500 
150 


$12,50 


500 


100 


3 

31250 

|50 


First,  \vhat  must  a  yard  of  cloth 
which  cost  166|5  be  sold  for,  to  gain 
50  per  cent.  ? 

Second,  If  a  yard  cost  1,66|,  and 
sell  for  250  cents,  what  is  the  gain 
per  cent.?  We  know  that  this  ought 
to  be  50.  We  say  then,  if  166J 
gain  83^,  what  will  100  gain  ? 


It  is  seen  here,  that  when  we  take  50  per 
cent,  from  250,  we  reduce  it  to  1 ,66| ;  and  that 
50  per  cent,  on  this  166|,  will  elevate  it  to  250 
again.  Some  think  it  singular  that  it  can  be 
done  in  this  case,  and  not  in  the  case  of  the 
first  example,  in  Variety  First.  The  great 
reason  of  this  difference,  is  the  difference  of 


PROFIT  AND  LOSS— VARIETY   THIRD.  77 

names.  In  the  latter  case,  we  are  falling  from 
the  amount  to  the  principal :  from  the  advance 
to  the  cost,  by  reducing  the  amount  of  per 
cent.,  to  its  par  or  first  value :  in  the  former, 
the  work  has  no  reference  to  finding  the  par 
or  cost  price ;  but  merely  to  laying  on  or  taking 
off  per  cent,  on  larger  or  smaller  sums;  as  10 
and  12  cents :  5  and  20  cents,  &c.  In  the 
latter  case  our  supposition  is  the  advance  per 
cent. :  in  the  former,  it  is  not  advance,  but  par 
per  cent.,  or  100  being  reduced  to  some  loss 
price.  Advance,  par,  loss,  reduced,  &c.,  be- 
come, therefore,  important  distinctions  if  we 
would  inquire  the  prime  reason  of  different  ope- 
rations, which  are  rather  apparently  the  same. 
Suppose  I  sell  a  yard  of  cloth  for  480  cents, 
and  thereby  lose  20  per  cent. :  It  is  evident 
that  I  have  sold  it  for  a  price,  20  per  cent,  too 
small.  Now,  many  would  think  that  we 
might  advance  the  480  cents,  20  per  cent., 
and  have  the  cost  price  ;  but  this  is  a  mistake : 
for  the  20  per  cent,  loss,  is  so  much  on  the 
cost  price,  and  could  not  be  calculated  on  the 
480 ;  because  it  is  the  reduced  price ;  and  20 
per  cent,  on  this  reduced  price  will  not  make 
as  much  as  on  the  cost  price.  We  say  there- 
fore, what  will  this  reduced  price,  480,  be 
advanced  to  for  nar,  if  80,  the  reduced  value  of 
per  cent.,  be  advanced  to  100,  par  ?  It  is  seen 
here,  that  we  have  reduced  opposite  reduced, 
and  last  on  the  right,  par  or  100  for  the  term 
of  answer.  We  find  that  the  cloth  cost  600 
cents,  thus, 


78 


RAINEY'S  IMPROVED  ABACUS. 


80 


480 
100 


cts.|600 


600 


100 


80  _ 

cts.|480 


600 


100 
120 


[20 


It  was  this  600  that  lost  the  20 
per  cent. ;  not  the  480  :  and  if  600  be 
reduced  20  per  cent,  by  Variety  first, 
the  reduced  or  loss  price  will  be 
found  480 ;  thus, 

This  may  be  proven  again,  by 
Variety  2d,  thus  :  If  cloth  that  cost 
600,  is  sold  for  480,  what  is  the  loss 
per  cent.  ?  We  know  that  it  was 
20  :  hence  20  per  cent. 


225 


500 

2 

100 


From  the  foregoing,  we  conclude,  that, 
To  find  the  cost  price,  when  an  article  has  been 
sold  at  a  specified  per  cent.,  gain  or  loss,  make 
the  selling  price  the  demand  ;  100,  increased  by 
the  gain  per  cent,  added,  or  diminished  by  the  loss 
per  cent,  subtracted,  the  same  name,  opposite; 
and  100,  the  term  of  answer,  on  the  right:  the 
answer  will  be  the  cost  price. 

If  a  man  sell  a  yard  of  cloth 
for  $5,  and  thereby  gain  12i  per 
cent.,  what  did  it  cost  him  ?  What 
will  500  advanced  price,  be  re- 
duced to,  if  112i  advanced  per 
cent.,  be  reduced  to  100? 

If  I  have  a  yard  of  cloth  that  cost 
4,441,  how  must  I  sell  it,  to  gain 
12i  per  cent.?  Variety  1st. 

If  I  buy  cloth  at  4441,  and  sell 
it  at  500  cents,  what  do  I  gain 
per  cent.  ?  Variety  2d.  It  is  evi- 
dent that  4441  gain  55 1  cents: 
then,  what  will  100  gain  ?  Here, 
12 1  is  the  answer. 


100 


14,441 

4000 
225 


4000 


|5,00 
100 


500 


PROFIT    AND    LOSS!     VARIETY    THIRD. 


79 


If,  when  wheat  is  sold  at  80 
cents  per  bushel,  20  per  cent,  is 
lost,  what  did  it  cost  ?  What  will 
the  reduced  price,  80,  be  advanced 
to,  if  80  reduced  per  cent,  be  ad- 
vanced to  100,  par? 

When  wheat  is  purchased  at 
100  cents,  and  sold  for  80,  what 
is  lost  per  cent.?  It  is  the  100 
cents  here,  that  lose  20 :  hence  20 
per  cent,  answer. 

If  hemp,  sold  at  $4i  per  cwt., 
gains  10  per  cent.,  what  did  it 
cost? 

If  hemp  cost  $4,09T1T  cents  per 
cwt,  for  how  much  will  it  be  sold 
to  gain  10  per  cent.? 

Purchased  flour  at  82,40  per  bar- 
rel, which  was  40  per  cent,  below 
cost :  what  was  the  cost? 


How  must  flour  that  cost  $4,00 
be  sold  to  lose  40  per  cent.? 


100 
100 


1104,50 
100 


4,09TV 


60  240 
100 


4,00 


100  400 
60 


2,40 


Sold  a  horse  for  120  dollars  and  thereby  lost 
20  per  cent.,  whereas  I  ought  have  gained  40 
per  cent.;  how  much  was  he  sold  under  his 
value?  It  is  plain  in  the  first  place,  that  if  I 
lost  20  per  cent,  in  selling  him  at  $120,  he 
must  have  cost  me  more  than  this :  conse- 
quently this  is  the  reduced  price.  It  may  be 
said,  therefore,  what  will  this  reduced  price  be 


80 


RAINEY'S  IMPROVED  ABACUS. 


£00,100 

140 


advanced  to,  for  cost,  if  80,  the  reduced  per 
cent.,  be  advanced  to  100?  This  will  give  the 
cost  price  of  the  horse.  Now,  this  cost  price 
must  be  advanced  40  per  cent.  Hence,  we 
say,  what  will  this  cost  price  be  advanced  to, 
if  100  be  advanced  to  140?  Thus,  the  two 
statements  are  combined, 

We  find  that  the  horse  should 
have  sold,  to  comply  with  these 
conditions,  for  210  dollars;  from 
which,  subtracting  120,  we  have 
a  loss  of  90  dollars.  We  might 
take  the  two  questions  separate- 
ly; or,  as  they  are  combined,  we 
might  drop  the  hundreds  for  mere  convenience, 
saying  80,  reduced  price,  may  be  advanced  to 
140,  selling  or  advanced  price. 

Suppose,  when  a  horse  is  sold  for  $120,  20 
per  cent,  is  gained,  whereas  a  loss  of  40  per 
cent,  might  be  sustained ;  how  much  is  he 
sold  over  his  value  ? 


210 
120 


90 


60 
60 


We  find  that  the  horse  might  have 
been  sold  for  $60,  and  is  consequent- 
ly sold  for  sixty  too  much,  in  making 
the  price  120. 

Again :  In  the  latter  example  the 
100  is  suspended  on  each  side,  while 
the  statement  is  still  quite  as  per- 
spicuous as  before. 

Operations  in  this  variety  of  profit  and  loss, 
are  quite  similar  to  those  of  discount.  In  dis- 
count the  advance  made  from  principal  or  par, 
to  amount,  is  based  on  the  losses  of  a  speci- 
fied time;  whereas,  in  this  case,  the  advance 


60 
60" 


PROFIT    AND    LOSS:    VARIETY    FOURTH.  81 

depends  on  a  usage  of  common  consent, 
whereby  an  individual  is  allowed  to  make  a 
reasonable  per  cent,  in  lieu  of  the  accommo- 
dation extended  to  those  around  him. 


FOURTH    VARIETY. 

If,  in  selling  a  yard  of  cloth  for  $5, 1  gain 
20  per  cent.,  what  will  be  gained  if  it  sell 
for  $6  ? 

In  this  case  we  may  say,  what  per  cent,  wild, 
be  gained  by  selling  at  $6,  if  by  selling  at  $5 
we  gain  20  per  cent?  Thus, 

It  would  gain  24  per  cent.  Now  5 
advancing  to  6  gains  1  dollar ;  what 
per  cent  is  it?  or,  what  will  100  gain 
if  5  gain  one? 


We   find   that   the   simple    advance 


124 
5100 


from  5  to  6  is  equivalent  to  20  per  cent. 
Here  the  general  advance  is  24,  and 
the  advance  from  5  to  6,  20  per  cent., 
which,  added,  make  44  per  cent,  as  the  gain 
in  selling  at  $6. 

We  say  above  first,  what  will  6  gain,  if  5 
gain  20:  and,  again,  what  will  100  gain,  if  5 
gain  1  ?  In  the  first  place  5  gains  20  as  a 
mere  addition  ;  and  in  the  second,  it  gains  1 
as  a  specified  per  cent.  Knowing,  then,  that 
we  must  yet  further  use  the  20  to  advance  it 
to  a  larger  amount,  and  likewise  the  100  to  get 
the  per  cent.,  we  combine  the  two  operations, 
by  adding  100  to  the  20  for  the  purpose  of 
getting  the  per  cent.;  suspending  one  of  the 
fives  on  the  left  ;  and  finally,  subtracting  this 


RAINEY'S  IMPROVED  ABACUS. 


100  thus  added  above,  which  leaves  the  true 
answer;  thus, 


6 
120—2 


144 
100 


44 


By  thus  adding  the  20  and  100, 
we  work  two  propositions,  and  per- 
form addition  at  the  same  time.  It 
is  very  seldom  necessary  to  add 
numbers  on  the  line ;  and  is  done  in 
this  instance  only  to  avoid  two  state- 


ments, and  to  shorten  the  work. 

When  one  price  is  advanced  to  another,  and 
a  gain  made,  we  subtract  100  from  the  an- 
swer, and  take  the  remainder  as  the  per  cent.; 
and  when  one  price  is  reduced  to  another,  the 
answer  is  subtracted  from  100,  while  the  re- 
mainder is,  as  before,  accounted  the  true  per 
cent. 

To  prove  this  answer  correct,  we  may  find 
the  cost  of  the  yard  of  cloth,  knowing  that  $5 
is  20  per  cent,  more  than  the  cost;  thus, 

We   find   by  this,  that   the 
cost  was  4£  dollars.     Now,  if 
4£  dollars,  in  advancing  to  6, 
gain   If   dollars,  what  is   the 
rate  per  cent,  gain  ? 

Thus,  by  variety  second,  we  find 
that  the  original  cost  was  4£,  and 
after  subtracting  this  from  the  sell- 
ing price,  by  variety  second,  find 
also  that  the  rate  is  44  per  cent., 


i 
11 


44 
as  above. 

In  selling  a  watch  for  $10,  I  lose  20  per 
cent ;  what  per  cent,  would  be  lost  in  selling 
it  at  $8? 


PROFIT    AND    LOSS:    VARIETY    FOURTH. 


83 


£08 


64 


Here  the  selling  price,  8,  is  the  de- 
mand; the  first  reduced  price,  10,  the 
same  name;  and  100,  reduced  by  the 
loss  per  cent,  subtracted,  the  same 
name.  It  may  be  observed  here,  that  if  the 
per  cent,  is  gain,  it  must  be  added,  and  if  loss,  sub- 
tracted. Taking  64  above,  from  100,  we  have 
36  per  cent,  loss,  for  the  answer. 

We  know  that  if  $10  lose  20  per  cent.,  8 
will  lose  16;  arid  likewise  in  descending  from 
10  to  8  we  lose  20  per  cent.:  now  20  and  16= 
36,  as  above. 

This  may  be  again  proven,  thus  :  What  will 
$10,  reduced  price,  be  advanced  to,  if  80  re- 
duced per  centum  be  advanced  to  100  ? 

Thus  the  cost  price  of  the  watch 
was  $12i  ;  now,  in  selling  for  8,  12i 
loses  4i;  hence  we  ask,  what  will  100 
lose? 


80*0 
100 

12* 


The  loss  in  reducing  to  this  price 
is  thus  found  to  be  36  per  cent., 
which  proves  the  work  correct  in 
every  part. 

If,  in  selling  a  pound  of  tobacco  for  30  cents, 
I  lose  10  per  cent.,  what  will  be  lost  if  it  is 
sold  at  25  cents  ? 


36 


25 
00—3 


The  result  is  75  cents,  which  we    £0 
subtract  from  100,  leaving   25  per 
cent,  as  the  answer.  75 

Now,  if  30  was  10  per  cent,  loss,  we  can 
easily  see  what  it  cost;  thus, 

3-4 


The  cost  was  33  J  cents. 


100 


84 


RAINEY'S  IMPROVED  ABACUS. 


If  33J  reduced 
100  lose? 

100  A  00 


to  25,  loses  8£,  what  will 


£ 

25_ 
25~ 


Again  the  question  is  proven  by 
finding  the  per  cent,  between  the  cost 
and  selling  price. 


3040 
130 


100 


If,  in  selling  a  horse  at  $30,  I  gain  30  per 
cent.,  what  will  I  gain  per  cent,  by  selling  him 
at  $40? 

We  subtract  100,  and  have  for  an- 
swer 73J  per  cent.  We  now  find,  by 
variety  third,  the  cost  of  the  horse, 
which  in  selling  at  $30  gained  30  per 
cent.;  saying,  what  will  30  advanced 
price  be  reduced  to,  if  130  advanced 
per  cent.,  be  reduced  to  100? 
130  30  Now,  23T^  being  the  cost  price,  we 

100      know  that  it  gains  16}|  in  advancing 
to  $40.  Therefore,  by  variety  second, 
^rV    we  say,  if  23T^  gain  16}f ,  what  is  the 
rate  per  cent.,  or  what  will  100  gain? 


Again,  the  work  is  proven  correct 
by  subsequent  statements. 


220 


From  the  foregoing  we  conclude,  that, 
To  find  ike  rate  per  cent.,  gain  or  loss,  when  an 
article,  sold  at  a  specified  price,  with  a  spe- 
cified gain  or  loss  per  cent.,  is  advanced  or  re- 
duced to  yet  another  price :  Make  the  last  selling 
price,  the  demand;  the  first  selling  price  the  same 
name',  and  100,  increased  by  the  gain  per  cent, 
added,  or  reduced  by  the  loss  per  cent,  subtracted^ 
the  term  of  answer.  If  a  per  cent,  is  gained^ 


PROFIT    AND    LOSS!    VARIETY    FIFTH  85 

subtract  100  from  the  answer;  if  lost,  subtract 
the  answer  from  100,  and  in  either  case  the  re- 
mainder  will  be  the  true  rate  per  cent.  Or, 

Proceed  according  to  directions  in  variety  third, 
and  Jind  the  original  cost ;  then  find  by  subtrac- 
tion the  difference  between  the  cost  and  the  selling 
price;  and  by  variety  second,  Jind  the  rate  per 
cent,  gain  or  loss. 

This  variety  of  profit  and  loss  is  of  but  lit- 
tle practical  value,  but  may  serve  to  awaken 
close  investigation  in  the  mind  of  the  reader. 


FIFTH    VARIETY. 

We  shall  consider  under  this  head  questions 
of  a  general  character,  with  particular  refer- 
ence to  the  combination  of  several  separate 
statements  into  one  general  statement,  by 
which  a  specific  answer  may  be  obtained.  As 
all  operations,  in  this  or  any  other  department 
of  profit  and  loss,  depend  primarily  on  pro- 
portion, it  will  be  necessary  to  make  all  of  the 
statements  occurring  in  combination,  by  the 
general  principles  indicated  in  simple  ratio. 
Such  statements  of  concatenated  proportions, 
might  be  called  conjoined  proportion,  which 
in  the  strictest  sense  is  but  making  the  answer 
of  a  preceding,  the  demand  of  a  subsequent  pro- 
portion. This  being  the  case,  and  all  propor- 
tion depending  in  solution  on  multiplication 
and  division,  we  conclude,  that  these  several 
multiplications  and  divisions  may  be  made  all 
together,  and  at  the  same  time. 

The  statement  of  questions  by  combination 


86  RAINEY'S  IMPROVED  ABACUS. 

gives  free  exercise  to  all  the  analytic  powers 
of  the  student's  mind,  and  tends  greatly  to  the 
cultivation  of  correct  modes  both  of  thinking 
and  reasoning.  It  will  be  found  necessary,  in 
all  such  questions,  to  commence  with  the  com- 
mencing point,  and  keep  up  all  of  the  natural 
relations  of  the  questions  until  the  term  of  an- 
swer is  found. 

Sent  to  Cincinnati  for  440  gallons  of  wine, 
and  paid  87i  cents  per  gallon :  paid  2  per 
cent,  commission  to  my  agents;  and  in  ex- 
changing specie  for  depreciated  Kentucky  pa- 
per, gained  10  per  cent-  premium:  lost  40  gal- 
lons by  leakage  :  how  much  must  the  remain- 
der be  sold  for  per  gallon,  to  gain  20  per  cent, 
on  the  prime  cost? 

It  is  evident  that  we  must  find  what  the  440 
gallons  cost,  at  the  price,  and  that  we  must 
then  increase  this  sum  2  per  cent,  for  commis- 
sion; for  the  commission  is  reckoned  on  the 
sum  of  money  paid  for  the  wine.  We  know 
that  this  depreciated  paper  will  be  received  in 
payment  for  the  wine  and  commission,  quite 
as  well,  if  necessary,  as  gold ;  and  that  $1 10  of 
the  paper  are  worth  $100  of  the  specie.  We, 
therefore,  find  what  sum  of  specie  will  pay 
for  this  paper,  or  cost  of  the  wine  and  com- 
mission. This,  then,  would  give  the  amount 
of  specie  to  be  sent  to  Cincinnati;  but  losing 
40  gallons,  we  divide  the  whole  quantity  of 
specie  paying  for  all  the  wine,  by  the  number 
of  gallons  left,  400,  and  find  the  advanced 
price  which  each  gallon  of  the  reduced  lot 
must  sell  at,  to  reproduce  the  amount  expen- 
ded. We  then  advance  this  price  20  per  cent., 


PROFIT    AND    LOSS!    COMBINATION.  87 

and  find  the  selling  price  of  the  wine,  per  gal- 
lon. It  is  a  settled  matter,  that  the  commis- 
sion must  be  paid  on  the  sum  invested,  wheth- 
er in  specie  or  paper.  It  is  likewise  easily 
seen,  that  it  is  the  amount  of  specie  sent  to 
Cincinnati  that  must  be  divided  by  the  re- 
duced quantity  of  wine ;  for  the  purchaser 
merely  wishes  to  know  the  advanced  price  per 
gallon  that  will  reinstate  him  in  his  expendi- 
ture, without  reference  to  the  10  per  cent,  dis- 
count saved ;  for  if  he  were  to  divide  the  price 
paid  in  paper,  he  would  reindemnify  his  loss 
of  40  gallons,  and  retain  also  the  10  per  cent. 
But,  as  it  is,  he  wishes  only  to  make  20  per 
cent,  on  the  whole  transaction,  by  advancing 
the  specie  price  of  each  gallon  of  the  reduced 
quantity  20  per  cent.  We  consequently  make 
the  statement  by  proportion ;  thus, 


What  will  440  gallons  cost,  if 
1  gallon  cost  87|;  what  will  this 
sum,  or  price  of  all  the  gallons,  be 
advanced  to,  if  100  be  advanced 
to  102,  for  amount  of  both  commis- 
sion and  payment?  Now,  what 
will  all  this  sum  of  money  in  pa- 


100 
110 
400 
100 


440 
175 

102 
100 
120 


1,07, 


per,  which  pays  for  the  wine,  be 
reduced  to,  for  specie,  if  1 10  paper,  opposite,  be 
worth  100  of  specie,  on  the  right?  Then, 
what  will  1  gallon  cost,  if  200,  the  remainder 
after  the  loss, be  worth  this  last  sum  in  specie? 
Here,  the  demand  1,  is  understood,  not  ex- 
pressed. Again :  What  will  this  price  per 
gallon  on  the  right,  as  demand,  be  advanced 
to,  if  100  opposite,  be  made  120?  We  find 


88 


RATNEY'S  IMPROVED  ABACUS. 


that  the  wine  must  be  sold  at  1  dollar  and  7T*y 
cents  per  gallon. 

The  several  successive  steps,  or  proportions 
in  this  statement,  may  be  made  separately,  as 
follows : 


1 

2 

440  gallons. 
175  cents  price. 

Per  ct.  100 

385,00  whole  price  of  wine. 
102  commission. 

Dig.        110 

392,70  amount  with  commission. 
100  specie. 

357,00  reduced  amount  of  specie. 

Red.  qu.  400 

1           1  gallon  demand. 
357,00  whole  price  in  specie. 

Per.  ct.  100 

89J       price  per  gallon,  red.  qu. 
120       20  per  cent,  profit. 

Ans. 

I,07y7  retail  price. 

The  concatenation  of  the  statement  is  here 
kept  up,  except  in  one  instance,  when  1  be- 
comes the  demand  on  the  right,  and  the  an- 
swer of  the  former  question,  35700,  is  the 
price  of  the  400  gallons.  We  might  dispense 
with  this  1 ;  and  use  it  here,  only  to  show  the 
full  proportional  relations.  Hence,  we  say, 
if  400  gallons  cost  $357,00,  what  will  1  gallon 
cost;  and  get  the  ratio  between  the  400  and 
the  1,  and  by  this  ratio,  which  is  ¥^,  multiply 
the  price  35700  cents,  and  thus  decrease  it  to 
89£  cents.  We  have  shown  the  absurdity  of 
attempting  to  multiply  together  two  denomi- 
nate things :  the  same  reasoning  is  true  with 
regard  to  dividing  one  denominate  by  another 


PROFIT    AND    LOSS*.     VARIETY    FIFTH. 


89 


denominate  thing,  as  cents  by  gallons,  or  gal- 
lons by  cents. 

Placing  the  110  on  the  left  according  to  dis- 
count may  seem  wrong,  until  we  reflect  that 
39270  is  the  amount  of  paper  which  pays  for 
the  wine;  and  which  we  wish  to  reduce  to 
gold.  This  amount  of  paper  must  be  com- 
pared with  another  amount  of  paper  that  will 
equal  per  centum  or  100  in  specie;  for  specie 
is  the  par  per  cent,  of  exchange ;  this  amount 
of  paper  we  know  is  110;  for  the  par,  100 
specie,  will  pay  for  this  sum  of  discounted 
funds.  Hence,  the  operation  is  one  of  pure 
discount. 

Bought  5  hogsheads  of  sugar,  containing 
each  1200  pounds,  at  2%  cents  per  lb.,  and  lose 
1000  Ibs.:  how  must  I  sell  the  remainder  per 
lb.  to  gain  6i  per  cent,  on  the  prime  cost? 

The  separate  statements  occur  in  the  follow- 
ing order:  How  many  pounds  will  5  hhds. 
make,  if  1  hhd.  make  1200?  what  will  these 
pounds  come  to,  if  1  pound  cost  2i  cents? 
then,  what  will  1  pound  cost,  if  the  5000 
pounds,  after  the  loss  of  1000,  cost  the  price 
indicated  by  the  last  answer  in  cents  ?  what 
will  this  advanced  price  be  yet  further  ad- 
vanced to,  if  100  be  advanced  to  106£;  thus, 


We  say  5  times  5  on 
the  right  make  25,  which 
goes  into  100  on  the  left 
4  times,  etc.,  etc.  In 
this  question  1  again  oc- 
curs as  demand,  while 
5000  is  the  same  name, 
and  the  preceding  an- 


1^00—3 


1651 


90 


RAINEY'S  IMPROVED  ABACUS. 


swer  in  cents,  the  denomination  of  the  an- 
swer; that  when  the  answer  is  obtained  it  may 
be  advanced  6i  per  cent  further. 

Suppose  I  purchase  90  yards  of  broadcloth 
at  $5  per  yard,  on  a  credit  of  1  year;  but  for 
ready  payment  am  allowed  a  discount  of  10 
per  cent.:  after  receiving  the  cloth  I  lose  10 
yards;  how  must  I  sell  the  remainder  per 
yard,  to  gain  10  per  cent,  on  the  prime  cost? 

What  will  90  yards  come  to,  if  1  yard  cost 
$5?  what  will  this  amount,  1  year  hence,  be 
reduced  to  for  ready  payment,  if  110  opposite 
be  reduced  to  100?  then,  if  this  price  pay  first 
for  90  yards,  or,  after  sustaining  a  loss,  for  80 
yards,  what  price  will  pay  for  1  yard?  what 
will  this  price  of  1  yard  be  advanced  to,  if  100 
be  advanced  to  110  for  selling  price?  thus, 

The  answer  5f  dollars,  or  $5,62i 
cents,  is  correct;  because  when  we 
deduct  from  the  cost  of  the  90  yards, 
10  per  cent  discount,  and  divide  the 
remainder  by  80  yards,  we  have 
the  cost  price  of  1  of  the  remaining 


190 
06 

80^00 


54 


yards ;  thus, 


1 

00 

110 

5 

80100 

te 

225 
352 


100 
44 

180 

10  per  ct. 


Now,  the  difference  between  this 
price  and  5f ,  for  which  it  sold,  is  |f|. 
We  may  now  prove  that  in  selling  at 
5f ,  10  per  cent,  is  gained,  by  variety 
second;  thus, 

If  5/f  gain  |f  f ,  what  will  100 
or  per  cent,  gain?  This  10 
per  cent,  gain  is  made  on  the 
amount  of  money  invested,  as 
has  been  shown;  consequent- 


ly,  on  the  first  cost. 


£—,100 120 


PROFIT    AND    LOSS:    COMBINATION.  91 

Purchased  cloth  at  $3  per  yard,  but  being 
damaged,  I  was  allowed  a  deduction  of  20  per 
cent.;  for  what  must  I  sell  it  to  gain  20  per 
cent.? 

The  first  statement  evidently  is,  what  will 
$5  be  reduced  to,  if  100  be  reduced  to  80  :  the 
second,  what  will  this  price  be  advanced  to,  if 
100  be  advanced  to  120;  thus, 

To  some  minds  the  old  diffi-  400  A 
culty  is  here  again  presented,  of 
finding  the  reduced  price  at  20 
per  cent,  loss,  which  at  20  per 
cent,  gain,  will  not  produce  the 
former  price.  We  must  recollect  that  when 
$5  lose  20  per  cent.,  they  will  be  reduced  to  a 
number  which  will  not  at  20  per  cent,  gain  a 
sufficient  amount  to  advance  to  $5  again. 
Five  dollars  in  losing  20  per  cent,  are  reduced 
to  $4;  but  $4  to  gain  20  per  cent  would  ad- 
vance to  $4,80  cents  only.  The  defect,  20, 
arises  from  the  $4  being  too  small  to  gain  at 
20  per  cent,  the  sum  that  $5  would  gain.  The 
first  statement  of  the  question  above,  if 
wrought  separately,  wrould  give  $4,  the  reduced 
price  of  the  cloth;  hence,  when  we  increase 
this  answer  20  per  cent.,  we  make  it  $4j. 

I  purchase  another  piece  of  cloth,  on  which 
the  sum  made  was  20  per  cent.;  in  considera- 
tion of  damages  he  lets  me  have  it  at  cost 
price,  or  at  a  discount  of  20  per  cent.:  what 
should  I  get  for  it? 

Five  dollars  is  the  advanced  price;  hence, 
we  will,  by  variety  third,  compare  with  it  120, 
advanced  per  centum,  and  reduce  it  to  cost  or 
par  price,  by  the  par,  100,  on  the  right.'  Then, 


92  RAINEY'S  IMPROVED  ABACUS. 

by  another  combined  statement,  we  will  ad- 
vance it  20  per  cent,  for  selling  price ;  what 
will  the  selling  price  be  ?  The  20  per  cent, 
advance  must  be  reckoned,  like  all  other  pro- 
fits and  losses,  on  the  cost  price ;  now,  20  per 
cent,  discount  has  been  taken  off  for  the  pur- 
pose of  finding  cost  price ;  thus,  when  the  cost 
price  is  found,  we  must,  by  20  per  cent.,  ad- 
vance it  again  to  its  former  advanced  price, 
$5;  thus, 

The  difference  in  these  two  opera- 
tions is,  that  in  the  latter  case  we 
discount  from  the  advanced  to  find 
the  cost ;  whereas  in  the  former,  we 
reduced  the  cost,  by  simple  loss,  to  find 
the  reduced  price. 

We  will  now  give  a  few  solutions  on  gene- 
ral principles. 

A  purchases  500  pounds  of  sugar  at  6  cents 
per  lb.;  how  must  he  sell  it  per  Ib.  to  gain  $20 
on  the  whole  lot? 

It  is  manifest  that  it  is  necessary  to  find,  first, 
the  cost  of  the  sugar,  which  is  $30,00.  To 
this  we  must  add  20,  making  $50,  the  whole 
price  that  the  500  Ibs.  must  sell  for,  to  gain 
£00 1  I  $20.  Now,  what  will  1  pound 
£0,00  cost,  if  500  pounds  cost  $50; 

1A   .       thus, 
10  cts. 

A  miller  sold  a  quantity  of  rye  at  $1  per 
bushel,  and  gained  20  per  cent.;  soon  after,  he 
sold  of  the  same  to  the  amount  of  $37,50,  and 
gained  50  per  cent.;  how  many  bushels  were 
there  in  the  last  parcel,  and  at  what  did  he 
sell  it  per  bushel  ? 


PROFIT    AND    LOSS!    COMBINATION. 


93 


150  37,50 
100 


$25 


2£,00 
3 


30  bu. 


We  know  that  Si  is  the  cost  price  of  one 
bushel,  with  20  per  cent,  profit  added  :  conse- 
quently, according  to  variety  third,  the  cost 
price  of  the  first  quantity  is  83i  cents;  thus, 

Now,  $37,50   is   50   per   cent.     120 
more  than  the  cost  price  of  the 
second  lot;  so,  by  the  same  pro- 
cess, we   ascertain   that  the   lot 
cost  $25;  thus, 

Now,  we  say,  how  many  bush- 
els will  $25,  the  cost  of  the  lot, 
buy,  if  83£,  cost  price,  buy  1  bush- 
el ;  thus, 

We  find  that  30  bushels  were 
sold.  Now,  if  these  30  bushels 
cost  $37,50,  what  will  1  bushel 
cost?  thus, 

It   is   found   by   the    following    30 
statement  that  he  sold  30  bushels 
at  $1,25  cents  per  bushel;  thus,         $  1,25  cts. 

We  see  here  beautifully  illustrated,  the  dis- 
tinction between  cost,  advanced,  and  reduced 
prices.  We  find  the  cost  of  each;  compare 
the  cost  of  1  with  the  cost  of  the  lot,  and  find 
the  number  of  bushels  in  each  lot.  Then, 
knowing  that  these  bushels  cost  the  price  of 
the  lot,  we  find  the  price  of  one  bushel.  This 
is  an  easy  and  simple  process  of  reasoning; 
yet  the  conditions  and  relations  of  such  ques- 
tions are  seldom  understood,  unless  nice  dis- 
tinctions are  made  in  the  terms. 

A  merchant  bought  a  parcel  of  cloth,  at  the 
rate  of  $1  for  2  yards,  of  which  he  sold  a  cer- 


1 
37,50 


94  RAINEY'S  IMPROVED  ABACUS. 

tain  quantity  at  the  rate  of  $3  for  5  yards; 
and  then  found  that  he  had  gained  as  much 
as  18  yards  cost;  how  many  yards  did  he  sell? 
We  know  that  the  cloth  cost  50  cents  per 
yard,  and  that  he  sold  it  for  60  cents;  conse- 
quently, he  gained  10  cents  per  yard.  Now, 
he  gained  as  much  in  selling  a  quantity  of  it 
as  18  yards  cost;  which  is  900  cents.  If, 
therefore,  10  cents  is  the  gain  of  1  yard,  of  how 
many  yards  is  900  cents  the  gain?  Nine  hun- 
dred is  the  demand;  10  cents  the  same  name, 
and  1  yard  the  term  of  answer;  thus, 


90yds, 

We  have  now  sufficiently  illustrated  all  of 
the  practical  operations  in  profit  and  loss,  to 
enable  the  careful  and  reflective  student  to 
perfect  his  knowledge  of  the  subject  by  exam- 
ples and  experiments  of  his  own,  both  in  the- 
ory and  practice,  in  all  of  the  usual  business 
transactions  of  life. 

A  great  number  of  questions  in  business 
come  under  the  head  of  variety  fifth,  which> 
judiciously  arranged  and  stated,  may  be  easily 
wrought;  and  frequently  with  one-fourth  the 
number  of  figures  required  by  former  methods, 
and  separate  statements. 

From  the  foregoing  illustrations,  we  deduce 
the  following 

SUMMARY    OF    DIRECTIONS. 

For  COMBINATION  OF  STATEMENTS  in  Profit  and 
Loss. 


DISCOUNT.  95 

I.  Place  first  on  the  right  the  quantity  of  the 
article  and  the  cost  price: 

II.  If  it  is  desired  to  advance  or   reduce  this 
entire  cost,  by  commission,  premium,  transporta- 
tion, drawback,  or  other  consideration,  place  100 
on  the  left,  and  100,  increased  or  reduced  by  the 
per  cent.,  on  the  right. 

III.  To  effect  a  discount,  place  100,  increased 
by  the  rate,  on  the  left,  and  100  on  the  right:     Or, 
if  profit  and  loss  follow,  place   100,  increased  or 
reduced  by  the  gain  or  loss  per  cent.,  on  the  right, 
in  the  place  of  the  100  or  par  of  discount. 

IV.  If  a  specified  portion  of  the  article  of  mer- 
chandise be  lost,  and  it  is  desired  to  know  how 
a  unit  of  the  quantity  must  be  sold,  subtract  the 
quantity  lost,  and  place  the  remainder  on  the  left. 

Or,  Make  the  whole  statement  a  concatenation  of 
proportions,  and  proceed  according  to  the  specified 
directions  in  the  various  rules  involved. 


DISCOUNT.* 

DISCOUNT  is  reckoned  by  two  methods  ;  one 
true;  the  other  false.  The  false  method  is 
very  often  used  by  business  men;  which  is 
merely  to  reckon  the  interest  on  the  amount, 


*  Discount  is  from  the  French  dccompte  to  count  back,  and  is  used 
synonymously  with  rebate,  which  is  from  rebattre,  to  strike  otV.  It  implies 
the  striking  of  a  portion  from  an  amount  made  of  separate  sums, 
as  is  the  case  in  discount,  where  the  amount  is  composed  of  the  original 
principal,  and  the  interest  supposed  to  have  accrued.  .Amount  is  from 
monter,  to  ascend  ;  which  is  from  the  root  of  the  Latin,  moiis,  a  moun- 
tain. 


96  RAINEY'S  IMPROVED   ABACUS. 

and  deduct  it  for  the  discount;  making  the 
remainder  the  present  worth.  By  the  true 
method,  the  amount  is  reduced  to  such  a  sum, 
as,  connected  with  its  own  interest  for  the 
time,  and  at  the  rate,  will  restore  the  amount. 
In  other  words,  the  interest  on  the  present 
worth,  which  is  equal  to  the  Discount,  will,  if 
added  to  the  present  worth,  restore  the  original 
amount  or  debt.  Some  individuals  use  both 
of  these  methods  :  the  former,  if  they  are  pay- 
ing out  money ;  the  latter,  if  receiving  it. 

The  theory  of  discount  is  based  on  the  sup- 
position, that  the  debt  becomes  due  at  a  future 
period,  and  bears  interest  from  date,  or  from 
the  specified  time  when  it  is  to  be  paid. 
Therefore,  it  is  necessary  to  use  some  standard 
of  present  and  future  value,  such  as  100,  or 
per  centum.  One  hundred  is  the  value  or 
standard  at  present  time;  but  100,  with  its 
own  interest  for  the  time  and  rate,  added, 
will  be  the  future  value,  of  such  standard. 
Suppose  the  time  1  year,  and  the  rate  10  per 
cent. :  the  standard  of  future  value  will  be 
110.  This  110  is  both  principal  and  interest, 
which  added,  make  the  amount  which  is 
always  the  future  value.  In  the  same  man- 
ner, the  debt  on  which  the  discount  is  to  be 
made,  is  both  principal  and  interest,  or 
amount:  hence,  the  propriety  of  comparing 
amount  of  debt  with  amount  of  standard,  and 
by  proportion,  reducing  the  amount  of  debt  to 
its  par  or  present  value.  In  the  supposition 
above,  if  110  dollars,  one  year  hence,  be 
reduced  to  100  dollars,  present  time  and  value, 
what  will  any  other  amount,  as  $100  debt,  be 


110 


100 
100 


THEORY  OF  DISCOUNT.  97 

reduced  to,  in  the  same  proportion :  that  is, 
if  $110  be  reduced  to  $100,  for  present  worth, 
what  will  100  dollars  be  reduced  to  for  present 
worth?  Here,  100  is  the  demand;  110  the 
same  name;  and  100  the  term  of  answer,  or 
present  worth.  We  state  accord- 
ingly, and  the  answer  will  be  the 
sum  of  money  payable  at  present 
time. 

By  annexing  two  ciphers,  this  answer  might 
be  obtained  in  cents.  The  present  worth  is 
901-0.  dollars;  or  90  dollars,  90  cents,  and  \±. 
This  sum  with  interest  for  one  year,  at  10 
per  cent.,  will  amount  to  100  dollars ;  which 
proves  the  position  correct,  that  the  interest  on 
the  present  worth  is  equal  to  the  sum  of  discount. 
It  is  necessary  to  make  a  distinction  between 
the  terms  used.  The  sum  is  the  whole  of  any- 
thing, from  summum,  the  whole :  as  the  sum  of 
interest,  the  sum  of  present  worth,  the  sum  of 
discount,  &c.  The  amount  is  the  result  of  two 
or  more  sums  added,  as  present  worth  and 
discount  added,  which  make  the  amount  of 
debt.  Amount  is  from  the  French  monter,  to 
ascend.  The  present  worth  is  the  portion  of 
the  debt  ^remaining,  after  the  discount  is 
deducted. 

It  is  necessary,  in  stating  this  sub-division 
of  numbers,  to  place  amount  opposite  amount, 
on  the  line,  that  we  may  ascertain  the  ratio 
between  such  different  amounts,  and  apply  it 
to  the  par,  present  worth,  or  100,  and  increase 
or  decrease  it  accordingly. 

What  will  be  the  present  worth  of  400  dol- 
lars, 10  years  hence,  at  6  per  cent  ?  Here,  as 


98  RAINEY'S  IMPROVED  ABACUS. 

in  all  other  cases  of  discount,  the  interest  on  1 
dollar,  for  the  time,  and  at  the  rate,  must  be 
ascertained,  and  added  to  100  cents;  and  the 
amount  placed  on  the  left  of  the  line.  One 
hundred  cents,  in  10  years,  at  6  per  cent., 
amounts  to  160  cents;  or,  100  dollars,  in  10 
years,  at  6  per  cent.,  will  amount  to  160  dolls. 
It  may  be  observed  here,  that  the  denomina- 
tion of  the  amount  on  the  left,  is  determined 
by  the  demand  on  the  left.  If  the  demand  is 
dollars,  the  amount  is  the  same;  and  if  cents, 
the  amount  is  cents;  the  left, or  amount, being 
merely  an  indenominate  standard. 

It  is  manifest  that  400  is  the  demand;  160 
the  same  name;  and  100  the  term  of  answer, 
,thus, 

400—25 


This  250  dolls. ,  in  10  years, 
at  6  per  cent.,  will  gain  150 
dollars,  which,  added  to  the 
250,  restores  the  original  amount,  400. 

What  is  the  present  worth  of  824  dollars, 
due  8  months  hence,  at  4i   per  cent.?     We 
ascertain  first  the  interest  on  1  dollar,  at  4J 
.  per  cent.,  thus, 

#_._i£  g >     |       We  now  add  this  3  cents 

##  to    100,  making    103,  which 

r^ — ; —    we  place  on  the  left  of  the 

3  cts"    '  line,  thus, 

The  answer  is  800  dollars, 
on  which,  in  8  months,  at  4-J 
per  cent.,  the  interest  would 
be  24  dollars.  This  added 
makes  the  amount  824.  When  it  is  necessary 
to  ascertain  the  sum  of  discount,  subtract  the 
present  worth  from  the  amount  due. 


100 


RULE  FOR  DISCOUNT.  99 

What  is  the  present  worth  of  $20,86,24, 
due  in  18  days,  at  6  per  cent,  discount?  We 
find  the  amount  of  100  cents, 
thus, 

The  amount  is  100,3.  This 
number  is  placed  on  the  left 
of  the  line,  thus,  100, 3 [20,86,24 

In  this  instance,  we  have  3  100 

mills    on   the   left,  after   the ^-Qnn 

cents,  and  on  the  right  four  «>|20,80,0 

hundredths  of  cents.  Having  this  one  decimal 
more  on  the  right,  than  on  the  left,  one  figure 
must  be  cut  off,  on  the  right,  for  hundredths. 
This  might  be  obviated,  by  using  another 
cipher  on  the  left,  making  3  mills  30  hun- 
dredths. In  such  case,  the  demand  and  same 
name,  would  be  of  the  same  denomination. 
From  the  foregoing  considerations,  we  are 
justified  in  making  the  following 

DIRECTIONS    FOR   DISCOUNT. 

To  find  the  present  worth  of  an  amount  of 
money  at  discount, 

Ascertain,  by  interest,  the  amount  of  100  dollar s^ 
or  cents,  for  the  time,  and  at  the  given  rate:  or 
ascertain  the  interest  on  one  dollar  for  the  time 
and  rate,  and  add  it  to  100;  make  the  amount  of 
the  debt  the  demand ;  the  amonnt  of  100  the  same 
name ;  and  100  the  term  of  answer:  the  answer 
will  be  the  present  worth.  To  ascertain  tlie  Dis- 
count, subtract  the  present  worth  from  the  amount. 
If  the  number  of  decimal  places  in  the  amount  on 
the  rig/it  is  greater  than  in  the  amount  on  the  left, 
cut  off  such  surplus  in  the  answer.  Cancel  as  in 


100  RAINEY'S   IMPROVED  ABACUS. 

other  cases ;  or,  if  this  be  impracticable,  multiply 
and  divide. 

It  may  be  well  to  offer  a  lew  remarks  to 
those  who  suppose  that  Discount  and  Interest 
are  the  same;  or  who  think  that  deducting 
the  interest,  is  a  fair  method  of  discounting. 

The  true  discount  of  100  dollars,  for  ten 
years,  at  10  per  cent.,  would  be  50  dollars. 
But  the  discount  by  the  false  method,  of  sub- 
tracting'the  interest,  would  of  itself,  be  100 
dollars ;  leaving  nothing  for  present  worth. 
The  absurdity  of  this  may  be  better  seen,  by 
taking  the  discount  on  the  same  sum  for  20 
years,  at  10  per  cent. ;  in  which  case,  the  dis- 
count would  be  200  dollars:  so  that,  if  de- 
ducted, would  leave  the  holder  of  the  note  100 
dollars  in  debt  to  his  creditor,  by  receiving 
payment,  or  making  settlement :  whereas,  by 
the  correct  method,  the  discount  never  can  en- 
tirely consume  the  debt;  as  there  must  always 
be  a  present  value.  We  have  no  space  for 
descanting  farther  on  the  beautiful  theory  of 
this  sub-division  of  Arithmetic;  and  will  pro- 
ceed to  the  consideration  of  per  cent.,  under 
other  heads. 

In  BANK  DISCOUNT,  the  interest  is  reckoned 
on  the  face  of  the  note,  and  deducted:  the 
remainder  is  the  present  worth,  or  sum  drawn. 
This  is  allowing  a  greater  rate  per  cent,  than 
is  specified  in  the  note ;  and  on  what  plea,  in 
morals  and  justice,  I  am  unable  to  learn. 


GIVING  NOTES  IN  BANK.  101 


FACE  OF  NOTES  GIVEN    IN  BANK. 

It  frequently  becomes  necessary  to  find  the 
face  of  a  note  given  in  bank,  to  draw  a  spe- 
cific sum  of  money.  Suppose  the  rate  per 
cent,  discount  be  6;  then  $100  face  of  note 
will  give  94  dollars  ready  money.  Suppose  it 
is  desired  to  draw  4700  dollars.  We  say, 
therefore,  what  will  4700  dollars  ready  money 
be  advanced  to  for  face  of  note,  if  94  dollars 
ready  money  be  advanced  to  100  dollars,  face 
of  note?  thus,  * 

Now,  the  interest  at  6  per 
cent.,  for    1    year,    deducted    - 
from  this   5000  dollars,  will 
leave  4700  dollars,  the  sum  to  be  drawn. 

Suppose  the  rate  to  be  4  per  cent.,  and  it  is 
desired  to  draw  1800  dollars:  we  know  that 
96  dollars  will  be  the  sum  drawn  for  the  face 
100,  and  state  accordingly;  saying,  what  will 
1800  be  advanced  to,  if  96  be  advanced  to 


1.100—25 


Here,  the  factor  12 
into  96  eight  times, 
and  into  180,  fifteen 
times;  again,  4  into  8  twice,  and  into  100,  25 
times;  while  2  on  the  left  into  10  on  the  right, 
five  times;  which,  multiplied  thus,  5X15X25, 
makes  1875,  the  answer.  The  interest  on  this 
r-.im,  at  4  per  cent.,  is  75  dollars,  which 
cieducted,  leaves  1800,  the  face  of  the  note. 
Hence,  To  find  the  face  of  a  note  given  at  bank, 
to  draw  a  specific  sum. 

Deduct  the  interest  of  100, /or  the  time  and  at 


102  RAINEV'S   IMPROVED  ABACUS. 

the  rate,  from  100;  place  the  remainder  on  the 
left  for  the  supposition,  cash  drawn;  the  sum  to 
be  drawn  in  cash,  on  the  right,  for  the  demand; 
and  100,  face  of  note,  last  on  the  right,  for  the 
term  of  answer :  the  answer  will  be  the  face  of  the 
note. 

Bankers  generally  ascertain  the  face  of  the 
note,  (though  they  frequently  wish  to  avoid  it 
altogether)  by  a  species  of  approximation,  by 
casting  interest  on  the  sum,  and  subsequently 
on  each  separate  sum  of  interest,  till  the  result 
is  too  small  toj3e  noticed  further.  The  precise 
face  of  the  note  could  never  be  obtained  in 
this  way. 


COMMISSION,  BROKERAGE,  &c. 

Commission,  Insurance,  Brokerage,  Taxes, 
&c.,  are  wrought  by  Proportion,  and  in  their 
standards  of  value,  are  based  on  per  ce?itu?n. 
Transactions  of  this  kind  are  generally  made 
without  reference  to  time,  that  is,  the  time 
required  for  the  specified  transaction,  is  con- 
sidered a  unit. 

Commission  is  a  specified  sum  paid  per  100, 
for  the  purchase  and  sale  of  merchandize,  &c. 
The  rate  of  commission  varies  from  1  to  20 
per  cent.  The  sum  paid  for  commission  is 
called  bonus,  which  is  the  amount  of  reward 
for  the  trouble  incurred;  from  the  Latin, bonus, 
good. 

A.  sends  to  B.  500  dollars  worth  of  books, 


THEORY  OF  COMMISSION.  10$ 

to  be  sold  on  commission,  and  agrees  to  allow 
him  2i  per  cent,  commission,  what  sum  does 
B.  receive?  Five  hundred,  ,  M^bM 
the  sum, is  the  demand;  100, 
or  per  cent.,  the  same  name; 
and  2J,  the  rate  bonus,  the 
term  of  answer.  The  answer  is  consequently 
124  dollars. 

The  same  name  and  the  term  of  answer,  are 
conformed  in  their  denomination,  to  that  of 
the  demand  :  for  being  only  standards,  they 
have  no  specific  names,  and  may  become 
dollars  or  cents,  as  indicated  by  the  name  of 
the  demand. 

What  is  the  commission  on  750  dollars 
worth  of  wheat  at  3J  per  ct.?  ,  ,-^Q 

Here,  we  suspend  the  100  j  J15 

on    the    left,   and    cut    off  2  | filFv) — 

figures    for    decimals    of    a  | 

dollar  in  the  answer.  28,02-J 

What  it  the  commission  on  800  bushels 
of  wheat,  worth  60  cents  per  bushel,  at  6^  per 

OP-ttt 

J   T   '  ' ,  .     .  ,  100i$00 — 2 

In  this  instance,  we  place  I  '60 

the  number  of  bushels,  and  A  25 

the  price,  on  the  right,  which irscToo 

is  equivalent  to  multiplying 
them ;  and  which  makes  the  statement  the 
following :  what  will  all  the  cents  that  800 
bushels  cost,  pay  for  commission,  if  100  cents 
pay  6i  cents  commission?  The  100  might 
again  be  suspended  on  the  left,  and  two  more 
figures  cut  off  in  the  answer,  for  hundredths 
of  cents. 

A  factor  receives  708  dollars  and  75  cents, 


104  RAINEY'S   IMPROVED   ABACUS. 

and  is  required  to  purchase  iron  at  45  dollars 
per  ton ;  he  is  to  receive  5  per  cent,  commission 
on  the  money  paid :  how  much  iron  will  he 
purchase  ? 

The  demand  is  the  amount  of  money,  708,75 
cents,  and  the  same  name,  100  with  the  com- 
mission added,  or  105;  and  the  name  of 
answer,  100.  This  would  give  the  amount  to 
be  invested.  In  this  instance  it  would  be  im- 
proper to  charge  commission  on  the  whole 
sum  of  money,  that  is,  to  charge  commission 
on  the  commission  received.  The  question  is, 
what  sum  must  the  factor  invest  in  iron,  so 
that  the  commission  on  the  same,  would  make 
such  sum  amount  to  708,75,  the  original  amount 
of  capital.  We  know  that  for  every  100  dol- 
lars that  he  invests,  he  receives  $5  commission  : 
this  added  makes  105:  now,  this  $105  capital 
will  make  100  investment ;  and  we  say  accord- 
ingly, what  must  708,75  capital  be  reduced 
to  for  investment,  if  105  capital,  be  made  100 
investment  money  ?  thus, 


105 
45,00 


708,75 
100 


1 15  tons 


This  will  give  the  whole  num- 
ber of  cents  that  may  be  invest- 
ed :  hence,  we  say  again,  how 
many  tons  iron  will  all  these 
cents,  in  this  involved  or  implied 


answer,  buy,  if  4500  cents  opposite,  buy  1  ton  ? 
Here,  we  combine  the  two  statements  in  one, 
and  have  for  the  answer,  15  tons.  That  the 
commission  should  be  charged  on  the  sum 
invested  only,  may  be  better  illustrated  by  the 
following  contrast :  A.  sends  to  B.  $100  worth 
of  books  to  be  sold  on  commission  at  25  per 
cent. :  what  commission  does  B.  receive?  It 


BROKERAGE.  1 Q5 

is  manifest,  that  as  B.  has  the  trouble  of  selling 
the  whole  lot  of  books,  that  part  which  pays 
his  commission  as  well  as  the  other,  he  should 
receive  commission  on  the  entire  100  dollars 
worth.  The  commission  is  consequently,  $25. 
Again  : 

Suppose  A.  send  to  B.  $100  with  which  to 
purchase  books  :  what  is  the  commission  at 
25  per  cent.  ?  Here,  the  commission  being 
ready  at  hand,  the  agent  has  no  further  trouble 
than  to  deduct  his  commission,  and  invest  the 
remainder.  It  would  be  manifestly  unjust  for 
him  to  charge  commission  on  his  commission, 
with  which  he  had  invested  no  time.  We  will 
suppose  that  if  he  wished  to  purchase  $100 
worth  of  books,  he  would  necessarily  send  to 
B.  100  to  invest,  and  25  to  pay  commission, 
or  125  capital  to  make  100  investment.  If 
then,  $125  capital  make  $100  investment,  what 
investment  will  $100  capital,  make? 


1/100— 4 
1/100-20 


It  is  thus  ascertained  that 
B.  must  invest  $80  and  reckon 
his  commission  on  the  same  ; 
which,  at  25  per  cent.,  would 
be  $20 ;  consuming  the  entire  $100. 

The  difference  consists  in  the  medium  that 
the  factor  operated  on  ;  the  one,  ready  capital ; 
the  other,  merchandize,  which  must  first  be 
converted. 

BROKERAGE. 

Brokerage  is  but  another  form  of  commis- 
sion, in  which  the  factor  or  agent  operates  in 
monies,  stocks,  &c.  The  rates  of  brokerage 
vary  from  ^  to  10  per  cent. 


106  RAINEY'S  IMPROVED  ABACUS. 

Brokers  operate  in  two  ways :  by  keeping 
a  current  account  with  their  dealers,  in  which 
they  charge  the  various  sums  of  premium  due 
them ;  or  by  deducting  the  premium  from  the 
capital  before  investment,  as  in  the  case  of 
the  purchase  of  books  just  mentioned. 

What  will  be   the   premium   or  bonus   for 

purchasing  300  shares  Whitewater  canal  stock, 

worth  $50  per  share,  at  If  per  cent,  premium? 

\Ttfak         Here,    the    number    of    shares    is 

100  50       multiplied  by  the    price  per   share, 

#j5         10°  placed  opposite,  and  the  rate,  If, 

j— —     or  |-   last  on   the  right.     We  might 

$|250  again  suspend  the  100  on  the  left, 
and  cut  off  two  decimals  for  it  on  the  right. 
Either  method  may  be  used  with  ease  and 
safety. 

B.  has  200  shares  Illinois  Canal  stock,  which 
he  \vishes  to  sell  20  per  cent,  below  par,  and 
agrees  to  pay  to  his  broker  77F  per  cent.,  for 
effecting  the  sale :  stock  worth  $100  per  share. 
The  number  of  shares  is  here  mul- 
tiplied by  the  price,  and  the  whole 
sum  of  stock  reduced  20  per  cent.,  as 
in  profit  and  loss,  which  gives  the 
true  value  of  the  200  shares.  Then 
we  say,  what  premium  will  all  of 
these  dollars  give,  if  100  opposite  give  T7¥  of 
a  dollar?  Hence  the  answer  112  dollars. 
We  might  suspend  the  100,  and  the  T\,  and 
ascertain  the  discounted  value  of  the  200 
shares  stock,  only. 

If  my  broker  purchase  for  me  300  shares 
Railroad  stock  at  10  per  cent,  advance,  and 


£00 

,100400 


100 


80 


RULE  FOR  COMMISSION.  107 

charge  me  1   per  cent.,  brokerage  on  the  sum 
invested,  what  will  my  stock  cost  me  ? 


Here,  the  advance  is  added  to 


£00 


the  100  on  the  right;  whereas,  in  ,100,100 
the  question  above,  the  20  per  1,100110 
cent,  was  subtracted  from  the  same  101 

number.     The  one  per  cent,  added         $133  330 
to  the  100  on  the  right,  is  not  to 
ascertain  the  premium,  but  the  whole  price  to 
which  the  stock  is  advanced  ;  and  is  identical 
\vith  Variety  1st,  in  Profit  and  Loss. 

From  the  foregoing  considerations,  we  are 
justified  in  making  the  following 

SUMMARY    OF    DIRECTIONS, 

for  calculating  bonus  and  premium  on  Commission, 
Brokerage,  Stocks,  fyc. 

To  ascertain  bonus,  premium,  tyc.,  Place  the 
amount  of  money,  merchandise,  stock,  fyc.  on  the 
right,  for  the  ^demand:  100  on  the  left,  for  tJie 
same  name;  and  the  rate,  bonus  or  premium,  last 
on  the  right,  for  the  term  of  answer ;  the  answer 
will  be  in  the  denomination  of  the  amount :  or, 
suspend  the  100  on  the  left,  and  cut  off  two  figures 
in  the  result,  for  hundrcdths.  Again  : 

To  Jind  the  sum  of  money  to  be  invested,  after 
the  commission  is  deducted  from  a  given  amount, 
Proceed  as  in  discount:  and  make  the  amount 
the  demand:  100  with  the  per  cent,  commission 
or  brokerage  added,  the  same  name  ;  and  100  the 
term  of  answer.  The  answer  will  be  the  sum  to 
be  invested. 

To  ascertain  premium,  or  selling  price  of  stock, 
when  effected  by  gain  or  loss  on  par, 


108  RAINEY'S   IMPROVED  ABACUS. 

Place  the  number  of  shares,  and  the  prize  per 
share,  on  the  right:  100  opposite;  and  100  in- 
creased by  the  gain  per  cent.,  or,  reduced  by  the 
loss  per  cent.,  on  the  right:  then,  to  ascertain  the 
premium,  place  100  on  the  left,  and  the  rate  on 
the  right ;  or,  to  ascertain  the  selling  price  of  the 
stock,  place  the  100  on  the  left,  as  before,  and  100 
on  the  right,  increased  by  the  gain,  or  reduced  by 
the  loss  per  cent.  The  answer  will  be  the  price 
of  the  stock,  including  gain  or  loss,  and  brokerage. 


INSURANCE. 

Insurance  is  security  against  hazard  or  loss 
of  property  on  land  or  sea ,  and  is  usually  di- 
vided into  two  kinds :  Fire  and  Marine. 

Fire  Insurance  is  that  which  secures  build- 
ings and  other  property  on  land :  Marine, 
which  is  from  maris,  the  sea,  is  to  secure  vessels, 
boats,  cargoes,  &c.,  on  sea,  or  on  rivers,  lakes, 
&c.  The  propriety  of  insurance  is  found  in  the 
fact,  that  by  a  single  accident  an  individual 
may  lose  his  entire  property,  with  little  or  no 
hope  of  recovery :  whereas,  if  he  pay  a  small 
bonus  to  an  association,  his  losses  may  be 
restored  to  him,  by  the  association  paying  him 
from  a  large  common  stock  fund  ;  which  fund 
is  kept  up  by  the  bonus  paid  by  each  indivi- 
dual who  insures.  Reason  dictates  that  it  is 
better  to  pay  a  small  share  of  our  profits  for 
certain  safety,  than  saving  it,  to  be  continually 
subject  to  lose,  not  only  the  small  sum  so 
saved,  but  the  entire  capital  on  which  the  hope 
of  all  gain  is  based. 


THEORY  OF  INSURANCE.  109 

Property  is  insured  in  two  ways  :  by  corpo- 
rations, which  are  legalized  associations  of  in- 
dividuals, with  specific  powers  and  privileges, 
based  on  a  definite  amount  of  common  stock 
fund  :  and  by  individuals,  according  to  private 
contract.  Insurance  effected  in  the  latter  way, 
is  called,  "  out-door"  insurance ;  and  is  never 
so  safe  as  the  former,  except  when  the  insurer 
is  careful  not  to  insure  property  to  a  1  arger 
amount  than  his  personal  property  would  in- 
demnify, in  case  of  loss,  on  part  of  the  assured. 

The  instrument  of  writing  given  to  an  indi- 
vidual as  evidence  of  his  insurance,  is  called  a 
Policy :  and  the  sum  paid  by  such  individual 
for  insurance,  is  called  Premium.  The  latter 
is  always  a  certain  per  cent,  on  the  value  of 
the  property  insured.  The  agent  of  the  com- 
pany, or  the  individual  who  signs  this  policy 
or  contract,  is  called  an  underwriter. 

The  question  arises,  what  per  cent,  should 
be  paid  for  the  insurance  of  property?  This 
is  always  according  to  circumstances  ;  as  prop- 
erty is  more  or  less  subject  to  damage.  Hence, 
insurers  divide  property  into  hazardous,  not 
hazardous,  and  extra  hazardous ;  charging  dif- 
ferent rates,  according  to  the  degree  of  expo- 
sure. These  distinctions  refer  more  particularly 
to  fire  insurance  on  land;  as,  in  marine  in- 
surance, there  is  but  little  difference  in  degree 
of  hazard. 

1st:  A  stone  or  brick  building  situated  re- 
mote from  other  buildings,  is,  from  its  location, 
and  the  quality  of  material  of  which  construct- 
ed, less  liable  to  destruction  by  fire,  than  simi- 
lar buildings  otherwise  located ;  or  buildings 
8 


110  RAINEY'S  IMPROVED  ABACUS. 

of  a  different  material  in  the  same  place.  For 
such,  the  rate  of  insurance  would  be  low. 

2d  :  Buildings  in  blocks,  densely  surrounded, 
where  fire  may  communicate  from  one  to 
another,  and  constructed  of  more  destructible 
materials,  being  more  hazardous,  are  insured 
at  higher  rates. 

3d  :  Buildings  devoted  to  purposes  involving 
greater  danger,  such  as  chemical  manufacto- 
ries, drug  establishments,  foundries,  &c.,  being 
greatly  exposed,  pay  yet  higher,  and  extra 
rates.  In  some  cases  of  similar  nature,  it  is 
impossible  to  effect  insurance  at  any  rate. 

The  unit  of  time  for  insurance  on  property, 
is  one  year.  Vessels  and  their  cargoes  are 
usually  insured  for  the  voyage.  Coasting  ves- 
sels are  generally  insured  by  the  year ;  being 
less  liable  to  loss  than  out-sea  vessels.  Coast- 
ers are  insured  at  rates  varying  from  3  to  8 
per  cent. ;  the  difference  being  the  result  of  a 
state  of  peace  or  war  in  the  vicinity.  Whale 
ships  are  insured  by  the  voyage,  at  from  4  to 
10  per  cent.  Insurance  on  goods,  ships,  stores, 
manufactories,  chattels,  dwellings,  &c.,  varies 
from  i  to  2i  per  cent.,  per  annum,  acording  to 
exposure. 

The  amount  of  insurance  taken  on  property, 
is  always  less  than  the  property  is  worth  ;  and 
is  generally  not  above  f  of  its  assessed  value. 
This  leaves  still  a  partial  degree  of  risk  on  the 
owner,  as  well  as  on  the  insurance  company, 
which  keeps  him  awake  to  the  safety  of  his 
property.  But  were  the  whole  value  insured, 
or  an  amount  greater  than  this,  it  would  be  an 
inducement  to  some,  to  destroy  their  property, 


THEORY  OF  INSURANCE.  HI 

for  the  base  purpose  of  converting  it  into 
money,  or  of  realizing  a  small  profit  on  the 
capital  invested  in  it.  Thus,  it  becomes  ne- 
cessary to  insure  a  definite  amount,  which 
entire  amount  may  be  recovered,  if  it  is  shown 
that  such  an  amount  of  property  has  been  lost. 
But  if  less  than  this  amount  is  lost,  the  whole 
insurance  cannot  be  recovered ;  only  a  share 
proportional  to  the  damage  sustained.  For 
instance ;  if  I  insure  property  to  the  amount 
of  $1000,  and  lose  one  half  of  it,  I  cam  recover 
but  $500  insurance. 

If  by  fire  or  other  accident,  a  loss  occurs  to 
property,  and  does  not  exceed  5  per  cent.,  it 
is  sustained  by  the  owner;  otherwise,  he 
might  carelessly  consign  everything  to  wreck 
around  him,  knowing  that  the  company  would 
have  to  pay  for  repairs. 

It  is  supposed  by  some  who  have  little 
experience,  and  less  common  observation, 
that  if  insurance  companies  can  make  money 
by  taking  risks,  individuals  certainly  can  by 
running  risks;  but  nothing  is  more  illogical. 
A  company  with  a  large  common  fund,  may 
locate  their  risks  in  a  great  number  of  places, 
so  that  a  loss  in  one  place  will  be  more  than 
overbalanced  by  the  gains  in  another ;  for  it  is 
scarcely  reasonable  to  suppose  that  losses  will 
occur  simultaneously  in  a  great  number  of 
places,  and  on  the  particular  property  insured 
by  a  certain  company.  But  suppose  an  indi- 
vidual lose  all  his  property  in  one  place :  he 
is  not  likely  to  have  enough  in  another  to 
retrieve  this  loss  by  equal  gains.  And  suppose 
he  lose  all,  which  not  unfrequently  occurs,  he 


112  RAINEY'S  IMPROVED  ABACUS. 

is  prostrate ;  his  energies  tied,  whatever  they 
be ;  for  his  gain-producing  property  or  element 
is  gone.  Now,  as  before  said,  which  is  better, 
that  he  save  a  little  and  risk  the  loss  of  all,  or 
lose  a  little,  and  secure  the  safety  of  the 
balance  ?  Common  sense  would  pronounce 
the  former  insanity,  in  all  cases  of  considera- 
ble hazard. 

Let  the  economical  merchant  in  Upper 
Missouri  invest  his  whole  worth  in  a  stock  of 
goods  in«St.  Louis,  and  ship  them  to  Lexington 
without  insurance.  The  boat  is  old  and  pretty 
well  insured,  and  the  owner  would  be  willing 
to  lose  her,  if  by  her  sinking  he  would  be 
enabled  to  purchase  a  new  one;  he  sinks  her: 
or,  a  good  boat  may  be  shattered  on  a  destruc- 
tive stump,  or  burned:  his  goods  are  lost;  his 
money  gone  ;  and  too,  his  friends,  one  by  one, 
or  en  7nasse,  have  disappeared;  what  resource 
is  left  him  of  the  earnings  of  former  years  of 
toil,  but  the  peaceful  shades  of  undisturbed 
poverty!  He  was  too  eager  to  be  rich; 
whereas,  had  he  insured,  he  might,  on  the 
recovery  of  his  money,  have  reinstated  him- 
self in  business,  at  the  defiance  even  of 
accident. 

Insurance,  like  other  divisions  of  Arithmetic 
involving  per  cent.,  is  wrought  by  Simple  Pro- 
portion. One  hundred,  or  per  centum,  is  the 
given  sum  that  gives  a  specified  premium. 
Any  other  sum,  therefore,  will,  as  it  is  larger 
or  smaller  than  100,  gain  a  greater  or  less  sum 
of  premium.  For  instance : 


INSURANCE— VARIETY  FIRST. 


113 


VARIETY  FIRST. 

A  building  is  insured  at  the  valuation,  2000 
dollars,  at  li  per  cent,  premium.  Now,  it  is 
quite  evident  that  the  li  premium  is  gained 
by  100,  or  it  could  not  be  so  much  per  centum. 
The  question  may,  therefore,  be  stated  in 
proportion,  thus  :  What  premium  will  insure 
2000  dollars  worth  of  property,  if  li  dollars 
insure  100  dollars  worth  of  property  ?  The  de- 
mand is  2000,  the  same  name 
100,  and  the  term  of  answer 
li  premium.  The  answer 
will  be  the  premium  for  in- 
surance in  dollars.  The  premium  is  $25. 

Again  :  what  is  the  sum 
premium  for  insuring  a  coast- 
ing vessel  worth  3000  dolls., 
at  7i  per  cent.  ? 

What  premium  must  be 
paid  on  a  shipment  of  goods 
from  New  Orleans  to  Havre, 
worth  6,280,  at  2i  per  cent.? 

What  is  the  annual  insurance 


,10014000 — 5 

41 5 


A00I3000-5 


on  a  cotton 
factory  worth  80,000  dollars,  at  4  per  cent.? 

It  is  perceived  here,  that 
the  rate  is  placed  last  on  the 
right;  consequently,  the  an- 
swer is  in  premium.  If  the 
amount  insured  is  dollars, 


£0,000—2 


$|600 

the  answer  is 
dollars ;  if  cents,  the  answer  is  cents ;  because 
if  the  demand  is  dollars  or  cents,  we  place 
100  dollars  or  cents  opposite,  and  the  premium 
must  be  a  given  sum  on  100,  of  the  same 


114       RAINEY'S  IMPROVED  ABACUS. 

denomination.  We  might  obtain  the  same 
result  in  the  cases  above,  by  suspending  the  100 
on  the  left,  and  cutting  off  two  figures  at  the 
right  of  the  result  for  decimals  of  dollars  or 
cents,  as  the  case  might  be.  Thus,  in  the  last 
example  : 

Here,  we  cut  off  the  two 


]  #0,000— 2 


decimals  of  a  dollar,  which 
are  found  on  the  right  as  the 
consequence  of  suspending 
the  100  on  the  left.  But  when  the  rate  is  frac- 
tional, which  is  not  un frequently  the  case,  we 
must,  to  multiply  by  it  with  facility,  place  the 
denominator  on  the  left  of  the  line.  In  cases 
of  prime  numbers,  or  of  odd  cents  in  the  sum, 
it  may  be  well  to  drop  the  100,  and  place  on 
the  right,  the  sum  and  rate  only.  What  is 
the  premium  on  a  whaler  for  the  voyage, 
worth  9783  dollars  ? 

9?83         ,       The.  100  could  not  be  ea- 
l  sily  divided  by  in  this  case; 
hence,  it  is  dropped,  and  two 


'  figures  are  stricken  off  for  it 

in  the  answer.  In  fact,  it  may  be  said  to  be 
used  for  the  purpose  of  keeping  up  the  pro- 
portional statement,  more  than  anything  else. 
A.  wishes  to  purchase  a  coasting  vessel 
worth  8000  dollars,  and  insure  it  for  one  year 
at  7f  per  cent. :  what  sum  of  money  will  be 
necessary  to  pay  both  purchase  and  insurance  ? 
This  may  be  done  in  two  ways  :  first,  by  find- 
ing the  premium,  and  adding  it  to  the  cost  of 
the  vessel;  2d,  by  saying,  if  100,  cost  price, 
amount  to  107J  cost  and  premium,  what  will 
8000  cost,  amount  to,  with  premium? 


INSURANCE— STATEMENTS  COMBINED.         115 

We  find  that  the  cost  of 
this  vessel,  with  the  premium  41431' 

added,  amounts  to  8620  dol- liaForf" 

lars  :  hence  we  infer  that  the 
premium  is  620  dollars. 

Questions  such  as  the  following  frequently 
occur,  when  it  is  quite  convenient  and  simple 
to  combine  statements.  Purchased  18000  Ibs. 
of  cheese  at  5i  cents  per  lb.,  on  a  credit  of  1 
year;  but  for  ready  money  I  am  allowed  10 
per  cent,  discount.  I  ship  the  cheese  to  Nash- 
ville, and  pay  If  per  cent,  insurance:  what 
will  the  cheese  cost  me  in  Nashville,  exclusive 

of  freights?  4 11  $000— 3 

In  this  statement,  we  find 
the  number  of  cents  that  the 


400 


100 

305 


$1915,00 


whole  quantity  of  cheese 
comes  to,  by  multiplying  by 
o£  or  y  cents :  I  then  dis- 
count 10  per  cent.,  by  saying,  what  will  all 
these  cents  be  reduced  to  for  present  worth,  if 
110  be  reduced  to  100?  Having  now  the 
present  cost  of  the  cheese,  we  say,  what  will 
all  these  cents  be  advanced  to,  if  100  be  ad- 
vanced to  101 1  for  insurance,  and  find  that  my 
answer  is  915,00  dollars.  The  two  figures  are 
cut  off  for  cents,  because  the  price  was  in 
cents,  and  all  the  operations  have  been  per- 
formed on  cents.  Were  the  freight  in  this 
case  10,  15,  or  any  other  rate  per  cent.,  it 
might  be  added  to  the  101$.  making  111$, 
116$,  &c.,  which  would  be  placed  on  the  line 
as  above. 


116  RAINEY'S   IMPROVED  ABACUS. 


,  VARIETY    SECOND, 

In  insurance,  teaches  the  method  of  finding 
the  rate  per  cent,  at  which  an  insurance  is 
effected,  when  the  premium  and  sum  are 
given;  as  in  the  following:  Paid  60  dollars 
premium  for  the  insurance  of  a  building, 
valued  at  3000  dollars  :  what  was  the  rate  per 
cent.  ?  The  question  here  is,  what  will  be  the 
premium  on  100  dollars,  or  per  cent.,  if  on 
$3000  it  is  60  dollars?  Hence,  per  cent.,  or 
100,  is  the  demand,  3000  the  same  name,  and 
60  dollars  premium,  the  term  of  answer.  We 
wish  the  answer  in  premium,  thus, 

Jt  is  found  by.  this'  that  the 
ra^e  insured  at  is  2  per  cent., 
which,  may  be  proven  as 


, 
I  follows:  if  100  gains  2  pre- 

mium>  what  wil1  300°  gain  ? 
The  answer  is  here,  60 
dollars,  the  sum  of  premium 
first  ascertained.  This  ex- 
ample, proving  the  other,  is  wrought  according 
to  Variety  1st,  in  Insurance.  From  the  nature 
and  statement  of  the  former  question,  it  may 
be  inferred  that, 

To  Jind  the  rate  per  cent,  of  Insurance,  when 
the  value  of  the  property  and  premium  arc  gain, 
Make  100  the  demand:  the  sum  insured,  the  same 
name;  and  the  premium,  the  term  of  answer  :  the 
answer  will  be  the  rate  per  cent,  premium. 

If  a  man  pay  40  dollars  per  annum  for  the 
insurance  of  his  house,  worth  1200  dollars,, 
what  rate  per  cent,  does  it  cost  him? 


INSURANCE—VARIETY  THIRD. 

The  rate  is  3 J  per  cent. ;  for 
this  is  the  price  of  insurance 
for  100  dolls,  worth  of  property. 
This  question  could  be  proven 
in  the  same  manner  as  the  one  above. 


VARIETY    THIRD, 

Is  to  find  the  sum  insured,  when  the  premium 
and  rate  are  given.  Thus,  if  I  pay  90  dollars 
premium  at  3  per  cent.,  what  is  the  sum 
insured?  It  is  clear,  in  this  case,  that  3  dolls, 
premium,  require  100  dolls,  worth  of  property ; 
now,  therefore,  what  will  90  dollars  premium 
require?  We  consequently  make  the  whole 
premium,  the  90  dollars,  the  demand :  the  rate 
of  premium,  or  per  cent.,  the  same  name ;  and 
$100,  value  of  property  insured  by  the  3  per 
cent.,  the  term  of  answer.  The  answer  is 
found,  therefore,  in  amount  of  property,  thus, 

We  find  by  this,  that  the  -  ^ 3 

amount  insured  is  $3000; 
which  may  be  proven  by 
showing  that  the  premium 


100 


$|3000 


on  this  sum,  at  3  per  cent.,  is  90  dollars. 
Therefore, 

To  find  the  amount  of  property  insured,  when 
the  rate  and  premium  are  given,  make  the  whole 
sum  paid  for  the  premium,  the  demand :  the  rate 
per  hundred,  the  same  name;  and  100  the  term 
of  answer.  The  answer  will  be  the  value  of  the 
property  insured. 

An  importer  paid  $700  premium,  on  wines 
imported  from  Madeira  to  Cincinnati,  which 


118  RAINEY'S  IMPROVED  ABACUS. 

was  H  per  cent,  on  the  sum  insured ;  how 
much  did  he  insure  ? 

We  find  that  the  cargo  of  wines 
was  worth  $56000.     This  may  be 
100          proven  by  finding  the  premium  on 

this  sum  at  li  per  cent.,  by  Variety 

$|56000  lgt)  which  would  be  $700:  or  by 
ascertaining  the  rate  of  premium  by  Variety  2d. 

VARIETY    FOURTH, 

Is  somewhat  different.  By  it  we  ascertain 
what  sum  must  be  insured  on  a  specified 
amount  of  property,  to  recover  both  property 
and  premium  in  case  of  loss. 

A.  owns  a  coasting  vessel  worth  $8400, 
and  wishes  to  insure  at  4  per  cent.,  so  that  if 
lost,  he  may  recover  both  the  value  of  the 
vessel  and  the  premium  :  what  sum  must  he 
insure  on  ?  While  the  rate  of  premium  is  4 
per  cent.,  nothing  is  plainer  than  that  on  a 
policy  of  $100,  only  $96  worth  of  property  can 
be  insured.  Therefore,  if  $96  worth  of  prop- 
erty,  be  advanced  to  $100,  property  and  pre- 
mium, the  question  arises,  what  will  $8400 
worth  of  property  be  advanced  to  for  both  prop- 
erty and  premium  ?  8400  is  the  demand,  96 
the  same  name,  and  100  the  term  of  answer, 
thus, 

96i8400         One  hundred  here,  is  the  amount, 
100       properly  speaking,  of  the  property 

•  — j   and  premium  ;  therefore,  the  answer 

518750  I  wjn  ke  sucn  an  amount  as  must  be 
insured  on,  to  secure  both.  Hence  the  result, 
8750.  To  prove  this  correct,  we  will  find  that 
the  premium  on  this  sum  at  4  per  cent.,  is 


INSURANCE—  VARIETY  FOURTH.  HQ 

$350,  which  subtracted,  leaves  the  first  value 
of  the  vessel.     Therefore,  we  conclude,  that, 

To  find  a  sum  to  be  insured  that  will  secure 
both  property  and  premium,  in  case  of  loss,  make 
the  value  of  the  property  the  demand  :  one  hun- 
dred, reduced  by  the  rate  per  cent.,  the  same  name; 
and  100  the  term  of  answer.  The  answer  will  be 
the  value  of  the  property  after  the  premium  has 
been  reckoned  and  deducted. 

Suppose  I  send  to  California  an  adventure 
of  $19000  worth  of  goods,  at  an  insurance  of 
5  per  cent.  :  what  sum  must  be  insured  on, 
that  in  case  of  total  wreck,  I  may  gain  both  the 
original  value  of  the  goods  and  the  premium  ? 

Here,  as  in  the  case   above, 
95  is  the  sum  of  property  which 


requires  for  amount  of  property 


100     2 


$|  20,000 


and  premium  100  :  hence,  we 
say,  what  amount  of  both  will 
$19000  property  require  ?  Thus,  it  is  seen  that 
20000  dollars  worth  must  be  insured.  It  may  be 
proven  by  finding  that  the  premium  on  20,000 
at  5  per  cent.,  is  $1000.  This  variety  of  opera- 
tion is  precisely  identical  with  that  of  finding 
the  face  of  a  note  "given  at  bank,  to  draw  a 
specific  sum.  A  similar  process  is  used  to  find 
the  amount  to  be  collected  by  tax-gatherers, 
when  it  is  desired  to  pay  both  the  taxes  and 
the  commission  on  them. 

Life  Insurance  is  reckoned  as  other  insurance; 
being  a  stipulated  per  cent.,  according  to  the 
age  and  health  of  the  assured. 

To  find  the  amount  to  be  collected  on  taxes,  to 
pay  both  tax  and  commission,  proceed  as  in  Variety 
4th  in  commission^  as  above. 


120  RAINEY'S   IMPROVED  ABACUS. 


TOLLS. 

Tolls  are  charges  made  for  the  transportation 
of  merchandise,  &c.,  by  canals,  and  railroads  : 
and  are  generally  a  specific  sum  per  mile,  on 
the  1000  Ibs.,  or  the  number  of  bbls.,  number 
of  perches,  number  of  feet,  &c.,  &c.  What- 
ever becomes  the  standard  of  a  given  article, 
as  1  barrel,  1  cord,  100  feet,  1  perch,  1000 
pounds,  &c.,  must  be  placed  on  the  left  of  the 
line  in  working  the  question.  For  instance  ; 
what  will  be  the  tolls  on  14000  Ibs.  of  flour,  20 
miles,  at  14  mills  per  mile  on  every  1000  Ibs.? 
Here,  we  place  down  the  num- 


1000 


14000 


14 


ber  of  miles  first  on  the  right,  and 
multiply  it  by  the  number  of  Ibs., 
by  placing  the  latter  under  the  for- 
mer. The  20  times  14000  would 
be  the  whole  number  of  Ibs.  to  be  carried  1 
mile :  now,  supposing  this  answer  to  be  in- 
volved, we  say,  what  will  all  of  these  Ibs.  cost 
on  the  right,  if  1000  Ibs.  opposite,  cost  14 
mills  ?  Here,  the  answer  is  mills,  because  the 
last  on  the  right  is  mills.  It  would  be  in 
cents,  if  the  price  of  the  1000  Ibs.  were  in 
cents.  Hence,  one  figure  is  cut  off  for  mills, 
and  the  answer  is  $3,  92  cents,  and  no  mills. 
What  must  be  paid  for  the  transportation 
of  18700  Ibs.  bacon,  60  miles,  at  15  mills  per 
1000  Ibs.  ? 


CANAL   AND  RAILROAD  TOLLS. 

Here,  we  may  place  the  price 
in  mills,  and  the  answer  will  be 
mills;  or, $16,83, no  mills.  The 
same  result  might  be  obtained  by 
calling  the  15  mills  1$  cents,  and 
placing  it  down  as  |,  thus, 

Here,  the  answer  is  in  cents, 
and  we  cut  off  accordingly.  It 
is  optional  with  the  operator 
whether  he  place  the  price  in 
mills  or  cents  :  but  if  in  the  for- 


121 

60 

18700 
_J_5_ 

$116,83,0 


1000 


18700 


$|16,83 


mer,  he  must  cut  off,  in   all  cases,  one  figure 
at  the  right,  for  mills. 


What  are  the  tolls  on  800  bbls. 
flour,  80  miles,  at  li  cents  per 
barrel  ? 


£0—4 

800 

3 


$|960,00 


What  tolls  on  20000  Ibs.  salt, 
63  miles,  at  1  cent  per  mile  ? 


What  tolls  on  192000  shingles, 
at  5  mills  per  1000,  for  20  miles  ? 
The  answer  is  19  dollars  and 
20  cents. 


What  tolls  on  1200  cubic  feet 
undressed  stone,  40  miles,  at  4 
mills  per  perch,  of  25  cubic  feet. 


,1000 


63 

20,000 

1 


$|12,60 


1000 


£0 
192000 


1 


$|19,20 


40 

1200—4 

4 


$|7,68,0 


122  RAINEY'S   IMPROVED  ABACUS. 

|M0* 


1003000 


What  are  the  tolls  on  3000  cubic 
feet  of  lumber,  at  H  cents  per  100 
feet,  for  18  miles  ? 


40 

8 

3 


What  are  the  tolls  on  a  pile  of  wood,  200 
feet  long,  40  feet  wide,  and  8  feet  high,  for  60 
miles,  at  H  cents  per  100  feet? 

Here,  it  is  unnecessary  to  multi- 
ply together  the  3  dimensions  of  the 
pile  of  wood  :  while  the  placing 
of  the  several  numbers  on  the 
right,  will  indicate  the  same.  We 
1576  00  might  ca^  these  three  dimensions 
64000  feet  of  wood,  and  use,  on 
the  right,  only  60,  64000,  and  f ,  which  would 
produce  the  same  result. 

It  frequently  becomes  necessary  to  reckon 
tolls,  when  it  is  not  convenient  to  go  through 
with  tedious  multiplications  and  divisions : 
hence,  the  necessity  of  combining  several  op- 
erations, as  above,  in  one  statement.  From 
the  foregoing  we  conclude,  that, 

To  ascertain  tolls  on  merchandise,  Place  the 
distance  in  miles,  and  quantity  of  freight  in  Ibs., 
feet,  4*c->  with  the  price,  on  the  right ;  and  the 
standard  measure,  of  the  specified  article  of  freight, 
on  the  left:  that  is,  if  tolls  are  charged  per  1000 
Ibs., place  1000  on  the  left:  if  on  100  feet, place 
100  on  the  left,  fyc. 


ANALYSIS    OF    COMPOUND    PROPORTION.         123 


COMPOUND  PROPORTION. 

We  have  seen  that  a  simple  proportion  is  formed 
by  the  combination  of  two  equal  ratios  ;  hence,  a  Com- 
pound Proportion  is  formed  of  two  or  more  of  these 
simple  proportions,  instead  of  a  compound  and  simple, 
ratio,  as  is  said  by  some  authors.  In  some  cases,  as 
in  the  following  example,  the  compound  proportion  is 
formed  of  the  two  combined  ratios  of  the  causes,  and 
the  one  ratio  between  the  effects ;  but  when  there  are 
more  terms  in  the  effect  than  one,  it  is  found  necessa- 
ry to  ascertain  a  number  of  ratios,  as  well  in  the  ef- 
fects, as  in  the  causes;  so  that  in  such  case, 
the  compound  proportion  would  be  composed  of  two 
compound  ratios,  instead  of  one  compound  and  one 
simple. 

It  may  be  shown,  as  in  the  question  following,  that 
by  two  or  more  statements  in  simple  proportion,  we  may 
easily  ascertain  the  result  of  a  compound  question. 

If  4  men  in  8  days  mow  12  acres  of  grass,  how 
many  acres  will  8  men  mow  in  16  days?  The  first 
statement  is,  as  4  men  to  8  men,  so  are  12  acres  to  24 
acres:  the  second,  as  8  days  to  16  days,  so  24  acreg 
to  48  acres  Thus,  how  many  acres  will  8  men  mow, 
if  4  men  mow  12  acres? 


The  answer  is  24 


12_ 
24~ 


Again :  How  many  acres  will  16  days  work  give,  if 
8  days  work  give  24  acres  ? 

The  answer,  48  acres,  is  thus  obtained  by 
two  separate  and  easy  statements.  Here 
it  is  seen,  that  in  each  of  the  two  ratios,  we 


24 

48 


have  made  a  proportion;  which  proportion 

has  in  each  case  shown  an  increase  in  the  number  of 


124  RAINEY'S  IMPROVED  ABACUS. 

acres,  until  this  number  has  become  as  great  as  if 
originally  multiplied  by  the  two  ratios  combined,  which 
make  4;  thus,  4  :  8  gives  2,  and  8  :  16  gives  2,  and 
twice  2  are  4,  the  combined  ratios.  Now  the  12  acres 
multiplied  by  this  ratio  4,  becomes  48  acres,  as 
before. 

In  compound  proportion  there  are  always  five  terms 
given  to  find  the  sixth,  seven  to  find  an  eighth,  nine  to 
find  a  tenth,  eleven  to  find  a  twelfth,  thirteen  to  find 
a  fourteenth,  fifteen  to  find  a  sixteenth,  etc.,  etc. 

In  a  compound,  as  in  a  simple  proportion,  the  two 
mean  terms  are  always  equal  to  the  two  extremes.  Be- 
cause there  may  be  ten,  twelve,  or  more  terms  in  a  com- 
pound proportion  question,  is  no  reason  that  the  theory 
and  the  governing  principles  of  the  question  are  any 
the  less  the  doctrine  of  means  and  extremes,  than  if 
there  are  four  terms  only.  All  of  these  several  terms 
must  be  so  classified  that  they  can  be  united  in  four 
distinct  bodies ;  which  four  bodies  become  means  and 
extremes  of  proportion.  The  great  difficulty  with 
most  authors  on  arithmetic  has  been  to  systematize 
this  classification,  so  as  to  present  it  as  a  general 
truth.  This  classification  depends  on  the  relations  of 
cause  and  effect ;  and  although  many  years  since  a 
European  writer,  Dr.  Lardner,  discovered  that  these 
causes  and  effects  were  the  great  foci  in  the  state- 
ment of  such  questions,  yet  no  system  of  bringing 
them  together,  or  into  their  appropriate  connection 
and  place,  was  discovered,  by  which  their  use  could  be 
availed.  It  has  been  so  in  all  sciences.  Many  bril- 
liant practical  and  useful  axioms  have  been  discovered, 
that  have  lain  dormant  in  neglect,  merely  because  the 
modus  operandi  of  their  application  did  not  accom- 
pany the  discovery,  so  as  to  give  it  efficiency. 

We  shall,  therefore,  consider, 

FIRST — The  principles  involved  in  Cause  and 
Effect : 


THEORY  OF  CAUSE  AND  EFFECT.      125 

SECOND — The  application  of  these  principles;  a?id, 

THIRD — The  form  of  statement  according  to  the 
necessities  of  their  relation,  so  as  properly  to  avail  all 
the  benefits  of  cancelation  in  their  reduction. 

Causes  are  anything  that  involve  action^  or  imply 
capacity ;  and  which  in  their  action,  or  the  contents 
of  their  capacity,  produce  effects.  Action  always 
commences  in  a  life-giving  principle;  pursues  some 
regular  medium;  and  invariably  shows  some  effect 
when  it  ends.  Capacity,  or  Geometrical  extent,  in- 
stead of  producing  any  effect,  merely  exercises  an  in- 
fluence over  effects.  It  produces  nothing,  because  it 
has  no  action;  this  action  being  always  essential  to 
the  formation  of  an  object.  Geometrical  extent  per- 
tains to  containing,  circumscribing,  or  consuming  ei* 
fects;  and,  as  such,  is  not  an  active,  but  a  passive 
cause. 

The  principles  in  Cause  and  Effect  pertain  strictly 
to  matter ;  and,  as  such,  admit  of  no  subdivision. 
They  form  simply  the  two  categories  of  the  produc- 
ing and  the  produc-ed :  the  agent  and  the  object. 

The  medium  through  which  causes  operate  to  pro- 
duce effects  cannot  be  real  or  material;  has  no  parts, 
and  is  only  imaginary.  It  is  like  a  ray  of  light  shoot- 
ing through  the  aperture  of  a  window  of  a  darkened 
room,  and  leaving  a  brilliant  spot  upon  the  wall.  The 
bright  spot  is  the  effect ;  the  sun,  the  cause ;  the  space 
between  them  being  only  a  straight  line,  conceivable 
and  imaginary,  though  without  parts.  A  house  that 
has  been  built  by  a  man  is  the  effect,  while  the  man 
is  the  cause.  No  relation  exists  between  these  but 
the  builder  and  the  built.  It  may  be  urged  that  there 
are  instrumentalities  necessary  to  enable  the  cause  or 
agent  to  do  this  work  or  produce  this  effect.  A  pro- 
per analysis  will  show,  however,  that  these  instrumen- 
talities should  be  merged  into  their  prime  causes,  and 
become  part  of  them.  It  would  be  said,  that  in 
9 


12$  RAINEYS  IMPROVED  ABACUS. 

building  the  house,  tools  were  the  instruments ;  but  a 
little  reflection  will  suggest  that  these  are  used  only 
to  give  the  hands  more  efficiency,  and  facilitate  the 
execution  of  the  work.  So  the  truth  is  resolved  back, 
that  the  man  is  the  plastic  cause  that  conforms  and 
molds  to  his  own  taste;  and  the  house,  the  object 
made. 

Ancient  philosophers  divided  and  subdivided  these 
agencies  and  effects,  in  actual  and  material  operations, 
into  ten  or  twelve  different  categories;  but  these  are 
both  useless  and  unreasonable,  and  tend  greatly  to 
confusion ;  while  the  practical  fact  recurs,  that  all  ob- 
jects in  nature  are  directly,  either  causes  or  effects. 

Causes  implying  action  as  well  as  capacity,  and 
causes  definitively  speaking,  being  active  and  endowed 
with  life,  we  conclude,  that 

FIRST — All  animate  things  are  causes.  Hence  we 
say,  active  causes  are  men  and  animals.  When  we  say 
men,  we  include  the  whole  human  race — men,  women, 
and  children.  These  constitute  a  higher  order  of 
causes,  being  endowed  with  reason ;  and  may,  there- 
fore, be  called  intelligent  causes.  When  we  say  ani- 
mals, we  refer  to  every  living,  moving,  creeping,  fly- 
ing thing  that  Grod  has  made.  Such  may  be  classed 
among  active  irrational  causes. 

Capacity  being  considered  cause,  we  conclude  that 

SECOND — All  time  is  cause ;  that  is,  every  conceiv- 
able subdivision  of  time,  as  centuries,  generations, 
years,  months,  days,  hours,  minutes,  seconds.  Time 
within  itself  has  none  of  those  active  powers  that  can 
entitle  it  to  the  name  of  an  efficient  cause;  but  it 
seems  to  be  co -efficient,  from  the  fact  that  it  gives  ex- 
tent or  capacity  in  which  active  and  operative  causes 
may  produce  effects.  This-  capacity,  or  sphere  of  ac- 
tion, it  will  be  shown,  is  as  essential  to  the  existence 
of  the  effect  as  the  cause  itself;  though  in  a  secondary 
degree  in  point  of  production. 


THEORY  OF  CAUSE  AND  EFFECT.       127 

Geometrical  extent  is  not  unfrequently  the  bounds 
determining  the  extent  of  matter  consumed  or  used; 
and  hence  the  occupation  of  the  vacuum  or  limit,  de- 
pends on  its  prescribed  limits  Hence  we  conclude, 
that, 

THIRD — Geometrical  extent  is  sometimes  a  cause, 
and  sometimes  an  effect :  a  cause  when  no  more  osten- 
sible or  efficient  cause  is  found  in  the  question;  and 
an  effect  when  the  dimension  of  something  produced 
by  an  efficient  cause. 

For  instance:  it  is  a  cause  when,  as  in  making 
cloth,  etc.,  a  certain  number  of  yards  in  length  and  quar- 
ters in  width  consume  a  given  quantity  of  wool :  it  is  an 
effect  only  when  the  dimension  of  something  produced ; 
as  the  length,  width,  and  hight,  of  a  wall,  etc.,  built 
by  a  given  number  of  men,  etc.  In  the  latter  case,  it 
is  an  effect,  because  the  work  is  effected  by  active 
causes,  such  as  men.  As  the  dimensions  of  a  wall, 
this  extent  might  be  a  cause;  as  in  determining  the 
quantity  of  stone  consumed  in  its  construction. 

Anything  representing  active  endeavor  may  be  call- 
ed a  cause :  hence  we  conclude,  that, 

FOURTH — Capital  is  a  cause,  when  it  produces  in- 
terest ;  because  it  has  delegated  to  it  all  the  productive 
and  cumulative  capacities  of  active  individual  effort. 
Capital  may  be  said  to  be  a  cause  only  when  it  produces 
interest.  There  are,  however,  instances  in  which  a 
sum  of  money  becomes  a  cause;  but  only  on  similar 
grounds  to  that  of  gaining  interest,  as  the  representa- 
tive of  active  powers.  It  is  only  in  transferring  ac- 
tion to  the  object  that  the  object  becomes  a  cause. 
When  this  is  done  by  conventional  usage,  as  in  the 
case  of  interest,  where  the  •  laws  give  this  quality  to 
capital,  it  may  be  called  a  relative  cause. 

It  is  a  self-evident  truth  in  common  sense,  that  a 
certain  number  of  separate  causes  must  produce  the 


128  RAINEY'S  IMPROVED  ABACUS. 

same  number  of  separate  effects :  or  that  the  extent 
of  the  effect  is  greater  or  less,  as  the  cause  is  greater 
or  less ;  and  that  the  quality  of  the  effect  depends  on 
the  nature  of  the  cause ;  or  as  the  change  is,  in  the 
cause,  so  must  it  be  in  the  effect. 

That  causes  always  exist  before  effects  are  pro- 
duced ;  and  that  no  effect  can  be  produced  without  a 
commensurate  cause;  and  that  causes,  as  causes, 
always  produce  effects,  are  alike  self-evident  and 
reasonable. 

The  cause  occupying  one  position  and  the  effect  an- 
other, thus, 

CAUSE E  FFECT, 

and  the  space  between  them  being  an  imaginary  line, 
no  one  can  doubt  that  they  are  as  directly  opposite 
in  their  nature  and  relations  as  east  and  west ;  right 
and  left.  Nor  will  it  be  disputed,  that  if  from  the 
cause  we  pursue  a  given  direction  to  find  the  effect,  we 
must,  in  returning  from  the  effect  to  the  cause,  trace 
back  the  line  by  which  we  first  sought  the  effect. 

We  know  that  the  united  energies  of  any  number 
of  causes,  cannot  in  a  natural  operation  produce  more 
than  one  effect  at  a  time :  otherwise  it  would  be  ne- 
cessary for  them  to  operate  in  two  directions  at  the 
same  time,  which  would  be  impossible ;  for  as  unity  of 
cause  produces  one  thing,  so,  whatever  produces  more 
than  any  one  thing  at  the  same  time,  must  be  more 
than  one  train  of  causes.  But,  as  no  object  can  be 
encompassed  or  circumscribed  by  the  direct  operation 
of  any  one  other  object,  influence,  cause,  or  thing,  so 
no  one  cause,  or  agent  in  the  sum  of  cause,  can,  unaided 
by  another  agent,  produce  an  effect. 

Although  men  may  be  considere.d  one  of  the  most 
active  species  of  causes  constituting  the  sum  of  causa- 
tion, yet,  the  8  men,  above  would  never,  of  themselves, 
mow  12  acres  of  grass ;  for,  however  active  they  may  be 


THEORY  OF  CAUSE  AND  EFFECT.      129 

as  one  element  of  causation,  yet  their  powers  must  have 
some  sphere  of  action',  and  this  sphere  is  time.  So 
that  when  we  combine  the  energies  of  the  men  with 
the  capacity  of  the  time,  both  together  constituting  a 
community  of  endeavor,  are  enabled  to  encompass  or 
effect  the  desired  object.  Any  one  thing  by  itself  as 
a  unit* is  inoperative;  but  when  multiplied  into  some- 
thing  else,  cumulates  or  expands  in  the  ratio  of 
squares.  It  is  this  expansion  of  the  combined  ener- 
gies of  causes  that  enables  them  to  encompass  effects ; 
and  it  is  the  necessity  of  this,  which  prevents  any  one 
bare  agent  in  a  cause  producing  any  effect  by  itself. 
*  It  may  be  said  that  certain  chemicals  produce  pow- 
erful effects  by  their  lone  agency :  this,  however,  can- 
not prove  the  position,  that  any  one  essential  element 
of  cause  can  produce  an  effect ;  for  whenever  any  chem- 
ical effect  is  produced,  it  is  the  consequence  of  the  com- 
bination of  different  essential  elements,  within  the 
chemicals,  or  in  the  air,  which  is  the  medium  of 
operation.  Hence,  different  effects  are  produced  by 
the  combination  of  different  elements;  the  combined 
effort  being  the  parent  of  the  inception. 

From  the  foregoing  we  may  safely  conclude,  that 
Causes  are 

MEN, 
ANIMALS, 
TIME, 

CAPITAL,  or 
MEDIUM  : 
Or,  whatever  produces  an  effect. 

Every  problem  in  Compound  Proportion,  and  in 
Simple  Inverse  Proportion,  is  composed  of  its  terms 
of  supposition,  and  its  terms  of  demand;  and  every 
supposition  and  every  demand  has  in  each,  causes  and 
effects. 

These  causes  and  effects  we  will  endeavor  to  clas- 
sify, so  as  to  form  a  rational  and  philosophical  state- 


130  RAINEY'S  IMPROVED  ABACUS. 

ment,  which  will,  at  the  same  time  that  it  is  clear,  be 
unencumbered  with  those  varied  unmanageable  depen- 
dencies of  terms,  common  to  the  old  form  of  state- 
ment; while  it  will  be  general  in  its  applica- 
tion, and  susceptible  of  all  the  abbreviation  prac- 
ticable in  cancelation.  Some  of  the  old  works  have 
indeed  canceled  some  little  in  this  department  of  arith- 
metic ;  but  not  to  any  very  useful  extent,  by  reason  of 
having  no  system  of  arranging  all  of  the  terms  so  as 
to  assume  the  form  of  dividend  and  divisor.  This  is 
the  great  desideratum,  so  far  as  canceling  is  concern- 
ed, in  all  modern-  works  that  pretend  to  use  it,  in  any 
other,  as  well  as  in  this  department  of  numbers. 

The  student  has  to  analyze  and  state  a  question 
thoroughly,  before  he  can  determine  what  terms  are 
divisors  and  what  dividends.  He  can  have,  therefore, 
no  regular  or  unique  system  of  statement ;  and  is 
at  all  times  necessitated  to  hunt  up  dividends  and 
divisors  merely  as  such ;  whereas,  by  this  method,  he 
has  the  well-defined  landmarks  of  cause  and  effect, 
and  states  according  to  the  two  great  and  infallible 
directions  that  nature  has  given  him,  in  all  created 
things. 

In  the  following  problem  we  will  assign  to  causes 
and  effects  their  appropriate  places. 

If  4  men  in  8  days  mow  12  acres  of  grass,  how 
many  azres  will  8  men  mmv  in  16  days  ? 

We  write  the  supposition  first,  and  then  the  de- 
mand, and  draw  a  line  under  each ;  thus, 

men.    days,    acres.  men.   days,   acres. 

4    .,    8    ,      12       —       8    .     16    ..    0 


A  short  line  is  likewise  drawn  under  the  causes  both  in 
the  supposition  and  demand.  Here  4  men  and  8 
days  are  the  causes  in  the  supposition,  and  12  acres 
the  effect :  in  the  demand,  8  men  and  16  days  are  the 


APPLICATION    OF    CAUSE    AND    EFFECT.  1  fU 

causes,  while  a  cipher  or  blank  indicates  the  place  of 
acres,  which  is  the  term  in  which  the  answer  is 
demanded. 

We  now  draw  two  vertical  lines  near  each  other ; 
the  left  for  the  supposition  and  the  right  for  the  de- 
mand. On  the  left  side  of  each  line  we  place 
and  on  the  right,  effects ;  thus, 

CAUSE.  EFFECT.       CAUSE.  EFFECT 


Supposition*  Demand. 

The  left  being  the  line  of  supposition,  we  write 
supposition  under  it ;  and  the  right  being  the  line  of 
demand,  we  write  demand  under  it.  Now,  all  of  the 
causes  in  the  supposition  are  placed  on  the  left  side 
of  the  left  line,  and  all  of  the  effects  on  the  right. 
All  of  the  causes  in  the  demand  are  placed  on  the  left 
side  of  the  right  line,  and  their  effects  on  the  right. 
It  appears  in  this  question  that  we  have  no  effect  in 
the  demand ;  consequently,  a  cipher  is  placed  on  the 
right,  to  show  that  it  is  deficient,  and  that  the  answer 
is  required  in  this  term;  thus, 

,E        It  is  perceived  here,  that  the  cause 
12         8 1  6  ^a^  Pro(^uces  the  l^  acres,  is  divided 

o  -jJ      into  two  parts,  men  and   days;    but 

these  men  and  days,  are  placed  together 
on  the  same  side  of  the  line,  that  their  powers  may 
be  combined  by  multiplication ;  thus  rendering  them 
unique  as  a  cause.  The  same  is  the  case  with  the  two 
-causes  on  the  other  line.  Therefore,  we  may  multiply 
the  two  causes  into  one.,  in  each  separate  case.  Four 
times  8  in  the  first,  make  32,  combined  cause ;  and  8 
times  16  in  the  second,  make  128,  a  similar  combina- 
tion. Instead  of  placing,  as  above,  the  separate  fac~ 


•c. 


132  RAINEY'S  IMPROVED  ABACUS. 

tors  of  each  cause  on  the  line,  we  may  place  their 
product;  thus, 

32 '12  128  0  Here  it  is  perceived,  that  we 
have  three  terms  of  a  proportion 
_ given  to  find  the  fourth;  that  is, 
the  two  means  'and  one  extreme,  to  find  the  other 
extreme.  Now,  we  know  that  if  these  two  means 
be  multiplied  together  and  divided  by  the  given  ex- 
treme, the  other  extreme  will  be  found;  which,  too, 
will  be  the  required  answer.  The  question,  as  now 
stated,  appears  to  be,  as  the  cause,  32,  is  to  the  cause 
128,  so  is  the  effect,  12,  to  the  required  effect,  48 ;  or, 
as  the  first  cause  is  to  the  first  effect,  so  is  the  second 
cause  to  the  second  or  required  effect.* 

This  proportion  may  be  transposed  as  other  pro- 
portions, into  eight  different  forms  or  readings;  the 
causes  and  effects  being  the  terms.  When  either  the 
means  or  extremes  are  multiplied  together,  they  are  the 
product  of  one  cause  and  one  effect ;  and  this  product 
is  divided  by  the  given  cause,  to  find  the  required  ef- 
fect ;  or  by  the  given  effect,  to  find  the  required  cause. 

But  suppose  this  question  be  changed  and  proven, 
by  saying:  If  4  men  in  8  days  mow  12  acres  of  grass, 
in  how  many  days.wiB  8  men  mow  48  acres?  Here, 
4  men  and  8  days  are  causes  in  the  supposition,  and 
12  acres  the  effect :  8  men  and  blank  (0)  days  the 
causes  in  the  demand,  and  48  acres  the  effect ;  thus> 


12     8 
0 


48  We  perceive  that  it  takes  both  men 
and  days,  two  causes,  in  the  supposition, 
to  produce  the  effect,  12  acres ;  and  in- 
fer that  two  similar  causes  should  exist  in  the  de- 
mand, to  produce  the  one  effect,  48  acres.  In  the  de- 

*  We  do  not  allude  here  to  the  specific  ratio  between  the  causes  and  their 
effects  ;  we  only  refer  to  their  categorical  and  regular  bearings.  Strictly 
speaking,  there  is  no  ratio  between  cause  and  effect,  except  that  acciden- 
tal ratio  which  depends  upon  the  nature  ef  things,  in  their  unclassified  and 
irregular  state.  We  desire  to  discard  this  forced,  and  irregular,  and  cir- 
cumstantial connection  or  ratio,  and  place  proportion  on  its  true  basis,- 
$i»  between,  things  tkat  are  alike.. 


PROPORTION  AMONG  CAUSES  AND  EFFECTS.     133 

mand  only  one  of  these  causes  is  found ;  hence,  the 
cause  is  deficient;  that  is,  one  of  the  terms  of  the 
means  is  deficient;  and  a  member  that  is  necessary  to 
constitute  the  causation,  producing  the  effect,  48. 
Hence,  while  all  of  the  extremes  are  multiplied  to- 
gether for  a  dividend,  the  product  should  be  divided 
by  all  of  the  remaining  means  to  get  the  required 
mean.  We  know  that  16  days  is  the  answer  required. 
Now,  the  product  of  the  extremes  is  4x8x48=1536, 
which,  divided  by  the  product  of  the  means  12x8= 
96,  gives  16,  the  number  of  days  required.  Thus,  we 
might  drop  any  one  of  the  terms,  of  either  the  means 
or  extremes,  and  ascertain  it,  by  dividing  the  product 
of  the  other  two,  by  the  remaining  terms. 

The  deficient  term,  or  that  in  which  the  answer  is 
required,  is  always  either  a  cause  or  an  effect.  This 
deficiency  must  always  be  indicated  by  a  cipher.  It 
never  falls  on  either  side  of  the  left,  or  line  of 
supposition ;  because  the  supposition  must  in  all  cases 
be  perfect,  and  have,  consequently,  no  deficiency;  but 
it  is  always  found  on  either  the  right  or  left  side  of 
the  right  line,  as  the  term  wanting  may  be  an  effect  or 
a  cause;  for  this  demand  is  always  deficient;  and  it 
is  this  deficiency  which  makes  it  a  demand,  by  asking 
what  will  supply  it ;  while  the  operation  is  performed 
for  no  other  purpose  than  to  supply  it. 

To  be  certain  that  the  blank,  or  cipher  indicating 
the  deficiency,  is  located  correctly,  we  must  consider 
whether  the  answer  desired  is  a  cause  or  an  effect. 
If  a  cause,  the  blank  is  placed  under  the  head  of 
cause,  on  the  left  side  of  the  line ;  if  an  effect,  it  must 
be  placed  under  the  head  of  effect,  on  the  right  side 
of  the  right  line.  Or  we  may  count  the  number  of 
causes  on  each  line,  and  if  they  be  equal,  the  defi- 
ciency is  an  effect.  Again :  we  may  count  the  num- 
ber of  effects  on  each  line ;  if  they  are  not  equal,  the 
deficiency  is  found;  if  they  are  equal,  the  deficiency 
must  be  among  the  causes. 


134  RAINEY'S  IMPROVED  ABACUS. 

After  placing  all  of  the  terms  in  the  question  on 
the  two  lines,  according  to  the  directions  given  above, 
notice  whether  the  blank  falls  among  the  mean  terms ; 
that  is,  between  the  two  lines ;  or  whether  among  the 
extremes,  or  terms  outside  the  two  lines :  if  it  fall 
among  the  inner  terms,  all  of  the  inner  terms  must  be 
placed  on  the  left  side  of  the  vertical  line,  or  line  of 
ratio,  on  which  the  question  is  to  be  wrought:  if 
among  the  outer  terms,  all  of  the  outer  terms  must  be 
placed  on  the  left  of  this  line.  This  is  based  on  the 
principle  of  dividing  by  the  mean  or  extreme,  as  the 
one  or  the  other  may  be  deficient,  to  ascertain  the 
deficient  term.  Being  thus  stated,  the  question  may 
be  canceled  as  in  other  cases.  Let  us  now  recur  to 
the  first  question : 

If  4  men  in  8  days  mow  12  acres,  how  many  acres 
will  8  men  mow  in  16  days?  and  state  it  thus, 

Here  we  inclose  the  mean  terms  en- 


12        8 
16 


0 


tirely,  which  are  12,  8,  16,  that  they 
~—  may  be  clearly  distinguished  from  the 
extreme,  which  are  4  and  8.  The  blank  falling  on  the 
outside,  the  outer  terms  become  the  divisors,  and  the 
inner  terms  the  dividend,  and  are  placed  on  the 
right;  thus, 

^  14—3      I 

Eight  equals   8 ;  4  into  12,  3  times, 
and  3x1^=48  acres,  the  answer. 
|48  acres,  i 

We  now  resume  the  second  question,  which  will 
prove  this  correct. 

If  4  men  in  8  days  mow  12  acres  of  grass,  in  how 
many  days  will  8  men  mow  48  acres? 

Four  and  8  are  the  causes,  in  the  supposition,  and  12 
the  effect;  8  and  blank  the  causes,  in  the  demand,  and 
48  acres  the  effect.  After  stating  the  question,  thus, 


COMPOUND    PROPORTION. 


135 


4.8 


*s  seen  that  ^ne  blank  is  among  the 
means;  hence,  the  means,  12  and  8,  be- 
come the  divisors,  and  are  accordingly 
placed  on  the  left  ;  while  4,  8,  and  48,  the  extremes,  are 
placed  on  the  right  of  the  line,  where  the  dividend  ia 
always  placed  ;  thus, 

£14 

m 


16  da. 


! 


Days  being  the  deficient  term,  the  an- 
swer is  16  days. 


Again :  If  4  men  in  8  days  mow  12  acres,  how 
many  men  will  be  required  to  mow  48  acres  in  16 
days? 


12 


0 
16 


48 


8  men. 


Here,  again,  the  mean  terms  be- 
come the  divisors.  Hence,  8  men, 
the  answer. 

Again:    changing  the  supposition  and  demand, 
If  8  men  in  16  days  mow  48  acres,  how  many  acres 
will  4  men  mow  in  8  days  ? 


848 
16 


0 


4ST-1 


12  acres. 


In  this  instance,  one  of  the 
terms  of  supposition  is  dropped, 
which  makes  it  deficient;  it  becomes,  consequently, 
the  demand;  while  all  of  the  terms  of  the  former 
demand  are  supplied;  making  the  demand,  in  its 
turn,  the  supposition.  Hence,  the  reason  for  placing 
it  on  the  left  line.  The  answer  is  12  acres. 

Again:  If  8  men,  in  16  days,  mow  48  acres  of 
grass,  in  how  many  days  will  4  men  mow  12  acres? 
thus, 


136 


RAINEY'S    IMPROVED    ABACUS. 


848 


16 


12 


The  first  question  has  now  been 
proven  in  all  of  its  terms  ;  showing 


40- 


8  da. 


conclusively  the  similarity  between  the  two  causes,  and 
the  two  effects,  as  the  four  terms  of  a  proportion. 

Having  presented  the  foregoing  rationale  of  Com- 
pound Proportion,  we  will  now  state  and  solve  a  num- 
ber of  problems  involving  all  of  the  varieties  com- 
mon to  this  department  ;  that  in  them  the  reader  may 
have  a  sure  guide  to  the  solution  of  all  similar  ques- 
tions, occurring  either  in  theory  or  practice. 

If  4  men,  in  8  days  of  10  hours  long,  mow  40 
acres  of  grass,  in  how  many  days  of  12  hours  long 
will  15  men  mow  60  acres? 


440 
8 
10 


1560 
0 
12 


Here,  the  different  lengths  of  the 
days  are  expressed  by  the  hours,  as 


A*  4 


2|da. 


causes,  in  each  case.  Hours  have,  therefore,  assigned 
to  them  the  place  of  cause,  as  all  other  causes  of  time ; 
and  cooperate  with  the  men  and  days  to  produce  the 
effect,  by  extending  the  time  or  sphere  of  execution, 
and  defining  its  limits.  The  answer  is  2|  days. 

If  4  men  in  20  days  of  8  hours  long,  build  a  wall 
400  feet  long,  32  ft.  wide  and  5  ft.  high,  in  how  many 
days,  15  hours  long,  will  6  men  build  another  wall, 
300  ft.  long,  80  feet  wide,  and  30  feet  high? 

While  4,  2,  and  80  are  the  causes  in  the  supposi- 
tion, 400,  32,  and,  5  are  the  dimensions  of  the  effect, 
which,  properly  considered,  is  a  unit.  In  both  cases, 
supposition  and  demand,  these  dimensions  are  made 
the  effect,  and  we  proceed  with  the  work  as  hereinbe- 


COMPOUND  PROPORTION  IN  FRACTIONS.    137 


fore  described.  The  answer  being  required  in  days 
the  blank  comes  under  the  head  of  cause,  to  show  it. 
Hence  the  mean  terms  are  the  divisors,  and  the  an- 
swer is  a  term  of  cause,  80  days. 


20 

8 


400 
32" 


0 
15 


300 

80 

30 


Two  ciphers  equal  two  ciphers;  8 
times  4  equal  32 ;  6  times  5  equal  30 ; 
3  into  15,  5,  and  5  times  4  equal  20; 
the  answer  is  80  days.  Now,  let  us  I 
prove  this  question  by  using  this  answer,  and  drop- 
ping some  other  term ;  the  80  feet  wide,  for  instance. 
The  question  will  be  as  follows : 

If  4  men  in  20  days  of  8  hours  long,  build  a  wall, 
400  feet  long,  32  feet  wide,  and  5  feet  high,  how  wide 
will  that  wall  be,  the  length  being  300  feet  and  the 
hight  30  feet,  which  6  men,  in  80  days,  16  hours  long, 
will  build  ? 


:400 

>32 

85 


2032 


80 
16 


300 

0 

30 


Here  it  is  seen  that  the  blank 
comes  under  the  head  of  effect,  one 
of  the  dimensions  of  the  wall,  which 
causes  us  to  divide  by  the  extremes. 


300 


7*00 
32' 

A 


85ift.w. 


The  following  question  involving  fractions  may  be 
as  easily  disposed  of  as  the  others.  We  will  simply 
place  the  mixed  numbers  on  the  two  lines  of  supposi- 
tion and  demand,  without  reducing  them,  until  they 
are  brought  to  the  line  of  ratio;  when,  after  being  re- 
duced as  in  other  cases,  to  improper  fractions,  the  nu- 
merators will  occupy  just  such  position  as  if  they 
were  whole  numbers,  with  their  respective  denomi- 
nators opposite  them.  Thus,  the  question  will  be 


S    IMPROVED 


the  I'orm  of  whole  nunibtjrs,  avuiuij-p:  thi 
inorous  iractioiial  difficulties  which  usually  render  such 
•solutions  too  complex  to  be  interesting  to  the  general 
reader. 

If  5  men,  in  7-J-  days  of  12^  hours  long,  dig  a  ditch 
400  feet  long,  4|  feet  wide,  and  6£  feet  deep,  how 
many  men  will  be  required  in  20  days,  6§  hours 
long,  to  dig  another  ditch  640  feet  long,  3f  feet  wide, 
and  3i  feet  wide  ? 


500 
4f 

4 


0640 
203£ 


We  have  placed  all  of  the  inner 
terms  on  the  left,  and  ah1  of  the 
outer  terms  on  the  right,  without 
observing  any  particular  order  as 
to  which  should  be  placed  on  the 
line  first. 


Men. 


815 


If  6  men  in  10  days,  8  hours  long,  dig  a  ca- 
nal 200  yards  long,  80  yards  wide,  and  3  yards  deep, 
in  how  many  days  of  12  hours  long,  will  5  men  dig  an- 
other canal  300  yards  long,  40  yards  wide,  and  16 
yards  deep,  supposing  the  difficulty  or  density  of  the 
soil  to  be,  in  the  latter  case  to  the  former,  as  5  to  2  ? 

It  is  palpable,  that  2  degrees  density  of  soil  must 
be  placed,  among  the  effects  of  the  supposition,  to 
determine  the  extent  or  difficulty  of  the  effect  to  be 
produced;  while,  for  the  same  cause,  5  degrees  of 
density  are  placed  among  the  effects  of  the  demand. 
The  same  results  could  be  obtained,  if  2i  were  placed 
among  the  effects  of  the  demand,  or  f  among  those 
of  the  supposition  ;  in  this  instance,  however,  the  sym- 
metry of  the  terms  would  be  lost,  and  the  statement  is 
made  according  to  the  first  suggestion;  thus, 


COMPOUND  PROPORTION  IN  FRACTIONS.    139 


10 


200 

80 

3 

2 


5J300 
040 


12 


16 


5 

6 

12 

10 

0 

200 

300 

80 

40 

3 

16 

2 

5 

80  da. 

We    find    in    Compound    Proportion 
many  conditions  that  can  be  appended 
to    questions,    which    ordinarily   render 
them    apparently    very    complex;     but 
these  conditions  may  be  disposed  in  the  most  summa- 
ry and  natural  manner,  by  assigning  to  the  terms  repre- 
senting them,  their  proper  places  as  causes  and  effects. 

How  long  will  8100  Ibs.  of  bread  serve  a  garrison 
of  100  men,  if  900  Ibs.  are  consumed  in  50  days  by 
20  men?  If,  in  this  instance  as  in  all  others,  indi- 
cates the  supposition,  which  is  20  men,  60  days,  and 
900  Ibs.  of  bread ;  while  the  demand  is  100  men,  0 
days,  and  8100  Ibs.  We  state  as  in  other  cases;  re- 
membering that  whatever  the  language  in  which  a 
problem  is  stated,  whether  the  men  eat,  or  whether 
the  bread  is  consumed, — whether  the  demand  or  sup- 
position be  given  first,  we  must  transpose  the  ques- 
tion, until  it  is  brought  before  the  mind  in  its  proper 
bearings ;  and  then  state  according  to  common  sense. 
Some  would  think,  at  first  sight  of  this  question,  that 
because  the  pounds  of  bread  served  the  garrison,  these 
pounds  must  necessarily  be  the  cause,  and  the  garrison 
the  effect.  Such  must  consider,  and  ask  in  what 
term  the  greater  action  exists ;  whether  in  the  bread,  in 
supplying  or  being  eaten,  or  in  the  men,  by  eating  and 
consuming  it.  The  latter  implies  action;  while  the 
former  is  only  subject  to  action,  and  as  an  object  in 
its  passive  form,  becomes  an  effect, 


201900 

50' 


100  8100 
0 


000  50 

#00—9 

90  days. 


140 


RAINEY'S  IMPROVED  ABACUS. 


If  60  horses  in  20  days  consume  15  bushels  of  oats, 
how  many  horses  will  600  bushels  feed  120  days? 

Horses  become  the  deficient  term ;  hence  we  divide 
by  the  inner  terms,  and  find  that  400  horses  will  con- 
sume the  600  bushels  in  120  days. 


60 
20 


15 


01600 

120[ 


000—4 


400 


This  question,  as  well  as  the  fore- 
going, and  all  that  will  follow,  may  be 
proven  in  as  many  different  ways  as  there  are  differ- 
ent terms  given  in  the  supposition  and  demand. 

If  400  musquitoes,  in  30  nights,  15  hours  long, 
raise  on  an  animal  60,000  lumps,  ±  of  an  inch  in  diam- 
eter and  i  inch  high,  in  how  many  nights,  of  10  hours 
in  length,  will  800  musquitoes  be  required  to  produce 
50,000  lumps,  -fj  of  an  inch  in  diameter,  and  ^V  of 
an  inch  in  hight? 

Here,  the  number  of  lumps,  their  diameter,  and 
hight  are  the  effects,  in  each  case;  while  the  musqui- 
toes,  nights,  and  hours  are  the  causes.  We  state 
accordingly. 


400 
30 
15 


60,000 


0 

800 
10 


50,000 


800400 

1030 
60,00015 
14 
18 
50,000 


169 
201 


In  this  question  numerous  other 
conditions  might  be  added:  it 
might  be  said  that  the  warmth  of 
the  night,  in  the  one  case,  was  to 
that  of  the  other  as  a  given  ratio : 
also,  it  might  be  said,  that  a  cer-  16£  m. 

tain  ratio  of  difference  existed  between  the  two  sub- 
jects operated  on,  as  to  adhesiveness  of  material,  etc. 
All  of  these  conditions  would  be  legitimate  in  such 
questions,  at  least  in  theory. 

We  now  come  to  the  consideration  of  questions  in 
which  the  causes  that  produce  effects,  instead  of  being 


RELATIVE    CAUSES   IN    PROPORTION.  141 

active,  are  merely  passive;  with  an  efficiency  dele- 
gated, rather  by  the  attendants,  and  controlling  cir- 
cumstances and  customs  of  the  day,  than  any  inherent 
or  vital  principle  of  action. 

Capital  and  time  are  causes  in  the  production  of 
interest,  not  because  they  have  any  active  powers  that 
would  enable  them  to  encompass  or  accomplish  this 
object,  but  because  public  common  consent  grants  that 
capital  may,  by  representing  individual  effort,  be 
supposed  to  do  this;  and  this  supposition  or  permis- 
sion of  the  thing,  is  equivalent  to  the  act  or  fact, 
"  other  things  being  equal,"  so  far  as  matters  of  com- 
mon consent  are  concerned. 

If  I  lend  my  friend,  in  his  need  of  money,  $200,  for 
20  days,  and  he  agrees  to  render  me  a  similar  accom- 
modation when  necessary;  and  when  my  occasion  re- 
quires it,  he  has  only  $150,  how  long  must  I  use  this 
sum  to  remunerate  me  for  the  loan  of  the  $200  ? 

The  question  appears  to  be  this :  If  $200  in  20  days, 
render  1  accommodation,  in  how  many  days  will  $150 
render  1  accommodation?  The  200  and  20  are  the 
causes  in  the  first  case,  and  1  accommodation  the  ef- 
fect; while,  in  the  other,  150  and  0  days  are  the 
causes,  and,  as  before,  1  accommodation  the  effect. 
State  as  follows: 


200 
20 


150 
0 


Hence  the  answer, 


150  200 

20 


26|  days. 


The  pupil  will  notice,  that  in  the  foregoing  propo- 
sition the  effects  are  equal;  that  is,  the  effect  in  the 
supposition  and  demand  is  the  same,  or  common.  We 
shall  have  occasion  to  use  this  fact  in  our  remarks  on 
Inverse  Proportion;  for  it  is  easily  seen,  that  if  the 
one  effect  equals  the  other,  they  can  both,  as  terms,  be 
dispensed  with :  hence,  in  the  statement  of  the  ques- 
10 


142 


RAJNEY  S    IMPROVED    ABACUS. 


tion  without  them,  we  would  find  only  three  terms 
given  for  ascertaining  the  fourth;  and  this  would 
make  it  a  simple  proportion. 

Let  a  few  other  questions  be  now  presented,  in 
which  capital  is  cause,  and  which  will  teach  the  pupil 
the  very  important  method  of  finding  principal,  time, 
and  rate  in  interest.  We  will  find,  first,  the  interest 
on  $200  for  3  months,  at  $6  gain  on  the  $100,  or  6 
per  cent.;  and  here,  as  in  other  cases,  we  must  have  a 
supposition,  which,  if  not  given  in  the  question,  must 
be  taken  from  legally-established  custom.  We  say, 
therefore, 

If  $100  dollars,  in  12  months,  gain  $6  interest, 
what  interest  will  $200  gain  in  3  months? 

The  causes  in  the  supposition  are  $100  and  12 
months,  while  $6  interest  is  the  effect;  and  in  the  de- 
mand $200  and  3  months,  while  the  effect  is  wanting. 
State  as  follows : 


10016 

12 


200 


0 


We  find  the  sixth  term  $3,  which 
is  the  effect;    showing  that,  if  $100 


8. 


3  int. 


in  1  year  gain  $6  interest,  twice  this  sum,  in 
time,  will  gain  ^  of  6,  or  $3  interest. 


the 


TO    FIND    THE   PRINCIPAL   IN    INTEREST. 

Having  ascertained  the  interest,  let  us  now  find  the 
capital  that  in  3  months  would  produce  this  interest. 
To  do  this  we  make  the  following  statement : 

If  $100,  in  12  months,  gain  $6  interest,  what  sum 
of  capital  will  gain  $3  interest  in  3  months?  thus, 

1006  0 

12  3 

Our  answer  is  $200  capital,  which 
is  correct ;    because  at  the  given 


100 

& 


$  200  prin. 


TO    FIND    PRINCIPAL,    TIME,    AND    RATE.         143 

rate  per  cent,  it  has  just  gained,  in  the  time  specified, 
the  $3  interest.  Here,  it  has  become  necessary  to 
assume  the  standards  of  per  centum  and  per  annum 
in  interest,  for  supposition. 


TO   FIND    THE   TIME   IN    INTEREST. 

Changing  the  statement,  it  may  be  well  to  find  the 
time  in  which  $200,  at  6  per  cent.,  will  gain  $3  inter- 
est :  and  when  we  say  at  6  per  cent.,  we  assume  that 
$100,  in  12  months,  360  days,  or  1  year,  gains  this 
$6.  Thus,  the  statement : 

If  $100,  in  12  months,  gain  $6  interest,  in  what 
time  will  $200  gain  $3  interest  ?  thus, 

1006  013 


12 


200 


The  answer  is  3  months  time.  3  mos. 

This  answer  may  again  be  changed  so  as 


TO   FIND    THE   KATE    OF   INTEREST. 

It  is  only  necessary  now,  to  complete  this  series  of 
proofs  and  proportions,  to  ascertain  the  rate  at  which 
money  is  lent,  when  a  certain  sum  has  gained  another 
specific  sum  of  interest.  We  commenced  in  the  ques- 
tion, by  saying,  that  the  rate  per  cent,  was  6  ;  and 
have  used  this  rate  in  the  statement  of  the  supposi- 
tion with  which  it  is  connected.  It  is  now  desired 
to  find  this  6  per  cent.,  per  annum.  When  we  say  6 
per  cent.,  per  annum,  we  mean  6  on  the  one  hundred, 
for  or  by  the  one  year,  or  12  months.  It  cannot  be 
denied,  that  this  $6  was  gained  by  the  $100  in  one 
year,  or  12  months.  The  former  supposition,  "If 
100  in  12  gain  6,"  now  becomes  the  demand;  one  of 


144 


RAINEY'S  IMPROVED  ABACUS. 


its  terms,  the  6,  being  deficient.  The  former  demand, 
therefore,  now  becomes  the  supposition,  and  is  as 
follows  : 

If  $200,  in  3  months,  gain  $3  interest,  what  in- 
terest will  $100,  or  per  centum,  gain  in  12  months,  or 
per  annum? 

The  answer  will  certainly  be  the  sum  that  the  $100 
in  1  year  would  gain ;  that  is,  $6,  which  proves  it  6 
per  cent,  per  annum. 


20013 
•I 


10010 


a— 2 


6  per  ct. 

We  may  in  a  similar  manner  find  the  rate  at  which 
interest  has  been  gained  on  a  given  principal,  and  for 
any  given  time,  by  saying,  if  such  principal,  in  such 
time,  gain  so  much  interest,  what  interest  will  $100 
principal  gain  in  1  year,  12  months,  or  360  or  365 
days,  as  the  case  may  be ;  for,  if  the  time  in  the  sup- 
position be  in  years,  1  year  must  be  used  in  the  de- 
mand: if  in  months,  12  months  in  the  demand;  and 
if  in  days,  360  days.  For  example : 

If  $60,  in  120  days,  gain  $1|  interest,  what  is  the 
rate  per  cent,  per  annum ;  or,  what  will  $100  gain  in 
860  days?  Thus, 


60111 

120J 


100  0 

360 


00  ^00 

^0  £00—3—2 


In  the  question  preceding  this, 
the  reader  will  perceive,  that  al-  6  per  cent, 

though  this  is  a  compound  statement,  yet  the  suppo- 
sition and  demand  are  located,  as  in  simple  proportion, 
with  the  term  of  answer,  $3  interest,  last  on  the  right. 

This  is  not  an  inverse  statement,  because  the  an- 
swer is  required  in  interest,  which  is  an  effect. 

Again :  Suppose  I  have  a  debt  of  $40  to  pay  in  80 


COMPOUND  PROPORTION:  CAUSES  OF  CAPACITY.  145 


days,  and  have  no  means  of  getting  the  money  except 
by  lending  to  some  individual  a  sum  of  my  capital, 
large  enough  to  gain  this  sum  of  debt,  in  the  given 
time,  at  6  per  cent.;  how  much  capital  shall  I  thus 
put  on  interest,  for  the  80  days,  to  gain  the  $40  ?  The 
question  would  be  stated,  thus, 

If  $100,  in  360  days,  gain  $6  interest,  what  prin- 
cipal will  gain  $40  in  80  days  ? 


1006 
360 


0 

80 


40 


£0  #00—6 

0100 

40—5 


It  is  thus  seen,  that  the  sum  to  be 
negotiated  is  $3000,  which  may  be       $  3000  cap. 
easily  proven,  by  asking  the  interest,  at  6  per  cent., 
on  this  sum,  for  80  days,  which  would  be  found  $40. 

We  next  proceed  to  the  consideration  of  causes  of 
capacity,  properly  so  called.  It  is  known  that  the  di- 
mensions 8  feet  long,  4  feet  wide,  and  4  feet  high 
make  1  cord.  Now, 

If  4  feet  high,  4  feet  wide,  and  8  feet  long  make  1 
cord  of  wood,  how  many  cords  will  a  pile  400  feet  long, 
60  feet  wide,  and  16  feet  high  make? 

In  each  case  the  dimensions  are  the  causes,  and  the 
number  of  cords  which  they  make,  the  effect.  We 
state  accordingly, 


400 
60 
16 


0 


The  answer  is  3000  cords  of  wood. 


460 


3000  cor. 


Now,  suppose  we  wish  a  pile  to  be  made  large 
enough  to  make  a  certain  number  of  cords,  where  two 
of  its  dimensions  are  given ;  for  example, 

If  4  feet  wide,  4  feet  high,  and  8  feet  long  make 
1  cord,  how  high  must  a  pile  be,  whose  width  is  8  feet 
and  length  40  feet,  to  make  10  cords?  thus, 


146 


RAINEY  S    IMPROVED    ABACUS. 


010 

8 
40 


The   pile   must  be  4  feet  high ;  | 
which  may  be  easily  proven  by  sus-          4  feet  hi. 
pending  some  other  dimension,  or  finding  the  number 
of  cords  that  these  complete  dimensions  will  make. 

If  10  yards  of  cloth,  J  of  a  yard  wide,  consume  40 
Ibs.  of  wool  in  the  facture,  how  many  pounds  will  it  take 
to  make  20  yards,  f  of  a  yard  wide  V 

We  have  shown  that  no  one  cause,  by  itself,  can  pro- 
duce an  effect,  until  united  with  some  other  cause ;  we 
have  likewise  shown,  that  no  cause,  or  combination  of 
causes,  can  produce  more  than  one  effect.  Now,  al- 
lowing the  existence  of  these  truths,  it  would  be  ridic- 
ulous to  say  that  the  pounds  of  wool,  in  the  question 
above,  was  the  cause  of  producing  so  much  cloth, 
so,  and  so  wide.  But,  to  say  that  the  length 
and  width  in  yards  and  quarters,  disposed  of,  or  con- 
sumed a  given  number  of  pounds,  would  be  both  rea- 
sonable and  natural.  The  want  of  a  little  discrimi- 
nation and  reflection  here,  will  involve  the  student 
in  innumerable  absurdities ;  while,  on  the  other  hand, 
a  proper  and  intelligent  attention  to  the  relations  and 
operations  of  causes  and  effects,  cannot  fail  conduct- 
ing him  to  proper  and  palpable  results. 


1040 


200 


Our  answer,  93i  Ibs.,  is  the  effect 
of  the  capacity  of  the  20  yards  long 
and  £  wide.  These  two,  as  causes  of 


10140 


20 

7 


93i 


capacity,  consume  or  appropriate  this  number  of 
pounds  of  wool,  as  the  effect.  We  are  apt  to  fall  into 
the  same  unheeded  absurdity  in  questions  like  the 
following : 


COMPOUND  PROPORTION  :  CAUSES  OF  CAPACITY.  147 


If  the  transportation  of  20  tons  of  freight,  30 
miles,  cost  $60,  how  many  tons  can  be  carried  800 
miles  for  $250? 

Here  the  tons  and  distance  are  the  causes,  and  the 
price  the  effect:  thus, 


2060 
30 


0 
800 


250 


250 


The  answer  is  3|  tons.  3|  tons. 

The  same  result  may  be  obtained,  though  impro- 
perly, by  saying  that  the  dollars  are  the  causes,  and 
the  weight  and  distance  the  effect.  If  all  the  rela- 
tions of  the  more  palpable  operations  of  cause  and 
effect  have  but  one  tendency,  and  which  reversed,  dis- 
orders the  whole,  certainly,  because  in  other  cases 
these  laws  and  relations  are  not  so  easily  perceived,  is 
no  reason  that  they  are  not  founded  upon,  or  governed 
by,  the  same  fixed  and  eternaj  principles.  Nor  can 
these  principles  be  neglected*  or  perverted,  without 
consequent  disorder,  at  some  time,  and  in  some  place. 

If  $40  pay  for  hauling  16  cwt.  of  hemp  28  miles, 
how  many  miles  can  -56  cwt,  be  hauled  for  $200  ? 


16140 

28| 


0|200 
56| 


200 


40  miles. 

It  is  an  easy  matter,  in  any  question,  to  determine 
what  terms  have  such  a  connection  as  to  require  that 
their  energies  be  combined ;  while  it  is  not  less  easily 
determined  which  term  appears  the  great  focus  of  all 
these  operations.  The  former  are  causes,  while  the 
latter  is  an  effect. 

If  a  cistern  8  ft.  deep,  20  ft.  long,  and  3  feet  wide, 
hold  340  barrels,  how  many  barrels  will  be  contained 
in  one  28  ft.  deep,  60  ft.  long,  and  15  ft,  wide? 

The  barrels  are,  in  each  case,  the  effects. 


148 


RAINEY  S    IMPROVED    ABACUS. 


20 
3 


340 


28 
60 
15 


— 17 
(—7 


15 


It  may  be  observed  here,  that  in 
all  cases,  such  as  the  one  above,  if  17850  bis. 

inches  be  annexed  to,  or  connected  with,  the  feet,  they 
must  be  reduced  to  the  fraction  of  a  foot.  Three  feet 
4  inches  may  be  reduced  by  saying,  4  inches  are  T42 
or  ±  of  a  foot ;  this  fraction  added,  makes  4|  or  y  feet. 

If  2150  inches  make  1  bushel,  how  high  must  a 
crib  be,  to  contain  800  bushels,  which  is  200  inches 
long  and  86  inches  wide? 

In  this  question,  it  is  the  geometrical  extent  of  all 
the  sides  of  the  bushel  measure,  which  is  the  cause ; 
the  2150  being  the  product  of  the  three  dimensions. 
The  cube  root  of  this  number  would  be  one  of  the 
sides;  and  this,  placed  down  three  times,  on  the  left 
side  of  the  line  of  supposition,  would  constitute  the 
causes  producing  the  effect,  one  bushel.  As  it  is  un- 
necessary to  find  this  side  an9  use  it  three  times, 
which,  at  best,  could  only  reproduce  the  number  2150, 
we  merely  write  the  2150,  which  is  the  product  of 
these  sides,  and  produces  precisely  the  same  result. 
In  place  of  the  two  remaining  factors  constituting 
2150y  we  use  two  inverted  commas;  thus, 


2150 


200 

86 

0 


800 


— 5 


100  in.  hi. 


It  is  seen,  that  the  crib  should  be  100  inches  high. 
By  this  mode  of  operation,  either  the  contents,  or  any 
of  the  sides  of  a  crib,  body,  box,  etc.,  may  be  found. 

The  question  above  may  be  proven,  by  asking,  how 
many  bushels  a  crib  will  contain,  which  is  200  inches 
long,  86  inches  wide,  and  300  inches  deep.  Again: 

If  2150  inches  make  1  bushel,,  haw  long  must  a  bos 


COMPOUND  PROPORTION  :  CAUSES  OF  CAPACITY;  149 


be,  to  contain  1  bushel,  which  is  20  inches  wide  and 
15  inches  deep? 


21501 


0 

20 
15 


We  find  that  it  must  be 


3— 4 


643 


7i  in. 


inches  long.  One  bushel  is  the  common  effect  in  both 
the  supposition  and  demand.  We  see  by  this  calcu- 
lation, that  a  box  7£  in.  deep,  20  in.  long,  and  15  in. 
wide,  will  contain  one  bushel.  This  may  be  proven, 
by  multiplying  the  three  sides  to  a  continued  product. 
We  might, in  this  instance,  give  a  short,  but  somewhat 
mechanical,  rule  for  finding  any  remaining  side  of  any 
figure  of  6  sides,  when  the  contents  were  a  unit  of 
any  given  standard,  or  any  number  of  units  of  such 
standard,  as  1  bushel,  or  80  bushels;  1  gallon,  or  20 
gallons,  etc. 

According  to  the  foregoing,  we  may,  in  a  similar 
manner,  find  the.  dimensions  of  a  box  or  cistern  that 
will  hold  a  given  number  of  gallons,  if  the  standard  of 
unity  be  first  assumed,  and  used  as  a  supposition; 
thus: 

If  231  inches  make  a  wine  gallon,  how  long,  wide, 
or  deep  must  a  box  be,  to  hold  any  specific  number  of 
gallons,  if  it  be  of  a  given  width,  or  depth,  etc.? 

Beer  gallons  may  become  the  standard  of  measure, 
by  using  282,  instead  of  231  inches 

If  231  inches  make  1  wine  gallon,  how  long  must  a 
box  be,  to  contain  100  gallons,  which  is  15  in.  wide, 
and  11  in.  deep? 


231 


0 

15 
11 


100 


100—2 


140  in.  long. 

By  this,  it  is  seen,  that  the  box  must  be  140  inches 
long. 


150 


RAINEY  S    IMPROVED    ABACUS. 


If  2688  inches  make  1  dry  bushel,  how  wide  must 
a  body  be,  to  hold  100  bushels  of  coal,  which  is  120 
in.  long,  and  32  in.  deep  ? 

Here,  2688  is  the  product  of  the  three  sides  of  the 
dry  bushel ;  and,  consequently,  must  be  considered  as 
the  unit  of  cause,  that  produces  or  contains  the  1 
bushel.  We  state  as  in  the  similar  case  above. 


2688 


0 

120 
32 


100 


100 


70  in.  long. 

The  width  of  the  body  must  be  70  inches. 

We  know  that  1  inch  wide,  1  inch  thick,  and  3T8/F 
inches  long,  make  1  pound  "of  cast  iron.  Now,  these 
dimensions  are  the  causes  in  the  supposition,  and  1 
pound,  the  effect. 

We  may  take  the  3T\\,  or,  which  is  equivalent,  3|i 
solid  inches  to  represent  the  dimensions  of  the  quan- 
tity, and  say, 

If  3f  £  inches  make  1  Ib.  of  iron,  how  many  Ibs. 
will  a  bar  make,  that  is  24  in.  long,  4  in.  thick,  and  6 
in.  wide? 


240 


6 


This    3fi    can   be.  very   easily 
used  without  the  least  difficulty  in  150  Ib. 

the  fraction,  as  f  f ,  as  above. 

The  question  may  now  be  changed;  and  having  the 
width  and  thickness  of  the  bar,  let  us  see  how  long  it 
must  be  to  weigh  150  pounds. 

By  careful  attention  to  the  statement  and  solution 
of  this  question,  any  question  arising  as  to  the  length, 
width,  or  thickness  of  bars  of  iron,  window-weights, 
etc.,  etc.,  may  be  easily  solved.  It  frequently  be- 
comes necessary,  after  the  two  dimensions  and  weight 
are  given,  to  find  the  other  dimension,  which  is 


COMPOUND  PROPORTION  :  CAUSES  OF  CAPACITY.  151 


often  very  troublesome  to  practical  men;  especial- 
ly if  many  fractions  are  found  in  the  work.  The 
question  is, 

If  3f  i  inches  make  1  lb.,  how  long  must  a  bar  be, 
that  is  6  in.  wide,  and  4  in.  thick,  to  make  150  Ibs? 


150 


From  this  work  it  appears,  that          24  in.  long, 
the  bar  must  be  24 in.  long,  which  is  true;  because 
this  length  was  used  to  find  the  weight. 

If  3-f.j-  inches  make  1  lb.,  94  inches  will  make  25 
Ibs.  The  product  of  96  is  composed  of  8  inches 
long,  4  inches  wide,  and  3  inches  thick ;  therefore,  in 
stating  questions  of  this  nature,  it  may  be  said, 

If  96  inches  make  25  Ibs ;  or,  if  8  in.  long,  4  wide, 
and  3  thick  make  25  Ibs.,  how  long,  or  thick,  or  wide 
must  a  bar  be,  with  two  dimensions  given,  to  make 
any  specific  number  of  pounds? 

If  8  in.  long,  4  in.  wide,  and  3  in.  thick  make  25 
Ibs.,  how  long  must  a  window -weight  be,  to  make  25 
Ibs.,  which  is  4 in.  wide  and  1-J-in.  thick? 


25 


0 
4 

H 


25 


The  bar  is  16  in.  long.     Again :          16  in.  long 

If  8,  4,  and  3  make  25  Ibs.,  how  wide  must  a  bar 
be,  to  make  25  Ibs.,  which  is  12  in.  long  and  1-Jin. 
thick  ? 


25 


0 
12 


25 


152 


RAINEY  S   IMPROVED    ABACUS. 


This  mode  of  stating  such  ques- 
tions can  be  easily  used,  if  any  at- 
tention be  given  to  the  general 
principles. 

5-j-  in.  wide. 

All  mixed  numbers  must  be  reduced  to  improper 
fractions,  here,  as  elsewhere. 

We  will  now  solve  one  or  two  other  questions,  and 
close  this  article  by  endeavoring  to  show  how  such 
statements  may  be  made  on  one  line,  as  in  Simple 
Proportion. 

If  10  men,  in  15  days,  8  hours  long,  compose  a  book 
of  16  sheets,  36  pages  on  a  sheet,  25  lines  on  a  page, 
and  60  letters  in  a  line,  in  how  many  days,  10  hours 
long,  will  40  men  compose  another  book,  of  48  sheets, 
32  pages  on  a  sheet  50  lines  on  a  page,  and  90  let- 
ters in  a  line  ? 


10 
15 


16 
36 
25 

60 


4048 


0 
1050 


32 
50 
90 


$— ,24 


00 


The  answer  is  24  days.  24  days. 

There  is  a  novelty  in  the  statement  and  solution  of 
this  question,  arising  from  the  fact  that  it  can  be 
wrought  without-  the  use  of  one  figure  more  than  is 
necessary  to  state  it.  Instead  of  canceling,  as  above, 
we  may  have  said,  16  into  32,  twice ;  twice  9  into  36, 
twice;  and  twice  4,  on  the  left,  equals  8:  10,  equals 
10;  cipher,  equals  cipher;  25  into  50,  twice;  2  into  6, 
three  times;  3  into  15,  5  times;  five  into  10,  twice; 
and  2  into  48,  twenty -four  times ;  the  answer,  as  above. 


COMPOUND  RROPORTION:  SINGLE  STATEMENT.  153 


One  more  example  is  given,  to  illustrate  the  method 
of  stating  on  one  line. 

If  4  men  in  16  days,  12  hours  long,  compose  a  book 
of  14  sheets,  24  pages  on  a  sheet,  44  lines  on  a  page, 
and  40  letters  in  a  line,  in  how  many  days,  8  hours 
long,  will  12  men  compose  a  similar  work,  of  42  sheets, 
16  pages  on  a  sheet,  48  lints  on  a  page,  and  55  let- 
ters  in  a  line  ? 


14 
24 
1244 
40 


16 


12 

0 


42 
16 

48 
55 


Twelve  equals  12  ;  8  into  16,  twice, 
and  twice  4,  on  the  right,  make  8, 
which  goes  into  40,  five  times  ;  5  into 
55,  eleven;  11  into  44,  four  times;  4  24  da. 
into  16,  four  times;  4  into  24,  6  times;  6  into  42, 
seven  times  ;  7  into  14,  twice  ;  and  2  into  48,  twenty- 
four  times  ;  the  answer. 

This  question  may  be  stated  in  the  following  order  : 

In  how  many  days,  8  hours  long,  will  12  men  pro- 
duce a  certain  effect,  if  4  men  produce  the  same  in  16 
days,  12  hours  long? 

The  answer  is  desired  in  days  ;  a  cause.  Hence, 
this  term,  16  days,  is  placed  last  on  the  right  for  the 
term  of  answer;  while  the  causes,  which  cooperate 
with  it  to  produce  the  given  effect,  are  placed  on  the 
same  side  of  the  line,,  where  their  energies  can  be 
multiplied.  Now,  these  causes  being  necessarily  on 
the  right,  the  causes  in  the  demand  must  be  placed 
opposite  them,  on  the  left.  Hence,  in  all  inverse  ques- 
tions, the  demand  is  placed  on  the  left:  thus, 


Here,  if  no  effects  were  to 
be  considered,  the  answer 
would  certainly  be  in  days  :  8 
days.  Thus,  we  see,  that 
when  the  answer  desired,  is  a 


Men, 
Hours, 


12  4     Men, 
12  Hours, 
16  Days, 


8  days. 


154  RAINEY'S  IMPROVED  ABACUS. 

cause,  the  demand,  that  is,  the  causes  of  demand, 
must  be  placed  on  the  left ;  and  those  of  the  supposi- 
tion, opposite.  Now,  if  causes  and  effects  are  always 
directly  opposite  in  their  nature,  the  effects  in  the 
demand,  must  be  placed  on  the  right;  and  those  of 
the  supposition,  opposite ;  thus, 


Sheets,  1442  Sheets, 

Pages,  2416  Pages, 

Lines,  4448  Lines, 

Letters,  40  55  Letters, 


It  is  seen  here,  that  the 
effects  occupy  the  reverse 
side  of  the  line.  This  is, 
both  because  causes  and 
effects  are  opposite  in  their 
nature ;  and  because,  when 
an  answer,  as  the  one  above,  is  desired  in  an  effect, 
the  proportion  is  direct,  requiring  that  the  demand  be 
placed  on  the  right.  Thus,  3  is  the  answer ;  which, 
multiplied  with  the  8,  gives  24,  as  in  the  former  work. 
Each  term  is  placed  opposite  the  term  of  its  own 
kind,  to  get  the  ratio.  This  order  of  statement  is  re- 
versed, if  the  answer  is  required  in  an  effect.  In 
such  case,  all  of  the  causes,  in  the  demand,  are  placed 
on  the  right,  and  those  in  the  supposition,  opposite ; 
while  all  the  effects  are  placed  opposite  their  respec- 
tive causes. 

From  the  foregoing,  we  deduce  the  following 

SUMMARY    OF    DIRECTIONS    FOR     COMPOUND    PROPORTION. 

Separate  the  question  into  terms  of  Supposition  and 
Demand :  Ascertain  which  terms,  both  in  the  suppo- 
sition and  demand,  are  Causes,  and  which  are  Ef- 
fects :  Draw  two  vertical  lines  :  Place  all  of  the  terms 
of  supposition  on  the  left  line  ;  and  all  of  the  terms 
of  demand,  on  the  right  line :  Place  causes  on  the  left 
side  of  each  line,  and  effects  on  the  right. 

If  the  answer  be  required  in  a  cause,  place  a  ci- 
pher on  the  left  side  of  the  right  line ;  if  in  an  effect, 
place  a  cipher  on  the  right  side  of  the  right  line. 


INVERSE    PROPORTION.  155 

When  the  cipher  falls  between  the  two  lines,  make 
all  of  the  inner  terms  the  divisor,  and  all  of  the  outer 
terms  the  dividend :  when  it  falls  outside  of  the  two 
lines,  make  all  of  the  outer  terms  the  divisor,  and  all 
of  the  inner  terms  the  dividend*  The  causes  are 
men,  animals,  time,  capital,  and  medium. 

To  state  on  one  line : 

When  the  answer  is  desired  in  a  cause,  place  all 
of  the  causes  in  the  demand,  on  the  left ;  and  all  of 
the  causes  in  the  supposition,  on  the  right ;  with  the 
respective  effects  of  each,  opposite: 

When  the  answer  is  desired  in  an  effect,  plact  all 
of  the  causes  in  the  demand,  on  the,  right ;  and  all  of 
the  causes  in  the  supposition^  on  the  left ;  with  the  re- 
spective effects  of  each,  opposite. 


SIMPLE  PROPORTION  INVERSE. 

The  consideration  of  Compound  Proportion,  leads 
to  that  of  Simple  Inverse  Proportion ;  the  great 
leading  principles  of  both,  being  cause  and  effect.  We 
have  seen,  from  the  solution  of  one  or  two  questions 
in  compound  proportion,  that  the  effect  in  a  compound 
question  being  the  same,  or  a  unit,  in  both  supposi- 
tion and  demand,  there  are  but  three  remaining 
terms  to  use.  The  effect,  being  unity  in  each  case, 
we  say,  that  the  two  terms  are  common;  and  conclude 
that  the  problem,  as  properly  enounced,  is  composed 
of  causes  only.  It  is  evident,  that  if  all  of  the  terms 
are  causes,  and  that  as  some  one  of  them  must  be  the 
denomination  of  the  answer,  therefore,  this  answer  must, 
when  obtained,  be  a  cause.  And  here  exists  the  differ  - 

*  Recollect  that  the  divisor  is  always  placed  on  the  left' side  of  the  line 
on  which  the  question  is  wrought,  and  the  dividend,  on  the  right. 


156  RAINEY'S  IMPROVED  ABACUS. 

eace  between  direct  and  inverse  proportion.  From  what 
has  been  said  of  the  relations  of  cause  and  effect,  it 
is  evident,  that  as  causes  exist  before  their  effects,  so 
the  production  of  these  effects  is  a  direct  or  first  ope- 
ration of  the  cause ;  hence,  it  is  the  province  of  di- 
rect proportion  to  ascertain  effects.  If,  again,  causes 
and  effects  possess  an  opposite  nature,  it  is  reasonable 
to  suppose,  that  when  the  cause  is  demanded  as  an 
answer,  the  course  pursued  to  find  it,  will  be  a  retro- 
grade from  the  effect,  to  the  cause.  Hence,  the  direct 
course  is  broken,  and  reversed ;  and  the  line  of  opera- 
tion, which  the  cause  first  pursued  to  find  the  effect,  is 
retraced  from  the  effect  to  the  cause.  This  inver- 
sion, or  turning  around,  is  the  basis  of  the  theory  of 
Inverse  Proportion.  The  term  is  derived  from  the 
Latin,  inverto,  to  turn  back. 

We  may  always  know  that  a  question  is  inverse, 
if  there  are  no  given  terms  of  effect,  whatsoever,  above 
unity.  When  the  effect  is  not  common,  the  question 
is  necessarily  compound;  having  more  than  three 
terms. 

All  inverse  questions  are  properly  compound  propor- 
tion, and  may  be  solved  most  intelligibly  and  easily, 
by  a  statement  under  this  head,  according  to  the  great 
principles  just  treated.  Therefore,  when  they  are  re- 
duced to  the  form  of  simple  proportion,  we  may  rea- 
sonably expect  an  entire  change  in  the  statement,  from 
questions  ordinarily  occurring  in  this  department.  The 
following  illustration,  according  to  compound  propor- 
tion, may  be  given : 

How  many  men  will  be  required  to  do  a  piece  of 
work  in  15,  which  can  be  done  in  24  days,  by  5  men? 

The  enunciation  of  the  question  may  be  changed, 
thus : 

If  5  men  in  25  days  do  one  piece  of  work,  how  many 
men  will  be  required  to  do  one  other  piece  in  15  days? 


THEORY    OF    INVERSE    PROPORTION.  157 

The  piece  of  work  is  the  effect,  in  each  case;  a 
common  effect  -,  and  is  located  as  follows : 


24 


0 
15 


8  men. 


It  is  seen,  that  in  dividing  by  the 
inner  terms,  the  15  is  placed  on  the 
left.  This  term  would  be  the  demand  in  simple  pro- 
portion ;  and,  it  is  evident,  that  it  is  placed  on  the  left, 
that,  by  dividing  the  product  of  the  other  two  terms, 
the  fourth  term  may  be  found;  which,  linked  with  the 
15,  would  complete  the  proportion.  It  is  not,  however, 
placed  on  the  left  as  a  factor  only,  to  find  another 
factor;  but  that  its  ratio,  with  a  similar  term  on  the 
opposite,  may  be  ascertained.  The  5  men  is  the  de- 
nomination of  the  answer,  and  must,  consequently,  be 
placed  last  on  the  right.  Now,  this  5  men  does  not 
equal  in  value  the  term  of  supposition  opposite,  the  15 
days,  which  is  the  sole  connection  between  the  two 
similar  terms  in  direct  proportion;  but,  as  a  cause,  it 
is  combined  with  another  cause,  the  24  days,  to  pro- 
duce the  common  effect,  one  piece  of  work.  That 
these  two  causes  may  occupy  a  position  where  their 
energies  may  be  multiplied  together  and  cumulated, 
the  24  days  is  placed  on  the  right  of  the  line,  in  the 
place  ordinarily  assigned  to  the  demand,  and  just  over 
its  cooperative  term,  5  men;  for  causes  producing  a 
common  effect,  cannot  be  separated.  Now,  to  obtain 
the  ratio  between  this  24  days  in  the  supposition  and 
the  15  days  in  the  demand,  the  15  must  be  placed  op- 
posite. Hence,  by  the  necessity  of  purely  philosophi- 
cal principles,  the  demand  in  inverse  proportion  is  in- 
verted, or  placed  on  the  left.  This  is  the  only  satis- 
factory reason  why  the  demand  changes  its  place,  or 
is  inverted.  These  laws,  necessitating  more  than  one 
cause  to  produce  an  effect,  and  an  actual  connection 
of  the  various  terms  employed,  so  as  to  elicit  their 


158  RAINEY'S  IMPROVED  ABACUS. 

combined  energies,  in  their  creative  capacity  of  some 
effect,  have  been  so  clearly  elucidated  and  demonstra- 
ted in  the  discussion  of  first  principles,  as  to  leave  no 
doubt,  on  the  part  of  the  reader,  that  both  terms  in 
the  supposition  must  be  placed  on  the  right;  one,  as 
the  denomination  of  the  answer,  and  the  other  as  the 
term  to  be  compared  with  the  term  of  similar  name, 
in  the  demand,  on  the  left.  The  two  ones  may  be 
dropped  in  the  statement.  Again : 

If  15  men,  in  8  days,  do  a  piece  of  work,  how  many 
men  will  do  the  same  in  24  days  ?  thus, 

The  demand  is  placed  on  the  left ; 
the  same  name  opposite  it ;  and  the 
,r  term  of  answer,  last  on  the  right.   Let 

{  us  prove  this  again. 
If  8  men,  in  15  days,  do  a  piece  of  work,  how 
many  days  will  5  men  be  required  to  do  the  same  ? 


L    g 

**  *L g  While  men  are  compared  in  this  state- 

—  — —     ment,  days  become  the  term  of  answer. 

|24days.    ! 

It  is  sometimes  difficult  to  determine  what  is 
cause,  and  what  effect;  especially,  when  geometrical 
extent  is  a  cause.  Therefore,  for  the  benefit  of  those 
who  are  not  experienced  in  the  designation  of  causes, 
we  give  the  following  directions  for  stating  such  ques- 
tions correctly ;  and  which,  to  some  minds,  is  the  only 
means  of  determining  whether  a  proportion  is  direct 
or  inverse: 

If  the  answer  desired  should  be  larger  than  the 
same  term  in  the  question,  phice  on  the  left  the  smaller 
of  the  two  terms  to  be  compared  by  ratio :  if  the  an- 
swer required  be  smaller,  place  on  the  left,  for  the  de- 
mand, the  larger  of  the  two  terms  to  be  compared. 
This  is  what  is  called  by  the  old  schoolmen,  "  more 
giving  less,  and  less  giving  more;"  which  simply 


INVERSE    PROPORTION    IN    FRACTIONS.  159 

means,  if  the  one  cause  or  factor  in  a  question  be 
larger,  the  other  must  be  smaller  ;  and  vice  versa. 

Most  authors  have  written  about  the  necessity  of 
inverting  terms  in  inverse  ratio,  as  they  call  it  (which, 
in  truth  never  existed),  as  well  as  about  more  giving 
less,  and  less  giving  more:  but  none  of  them  have 
ever  explained  satisfactorily  the  reason  for  this  inver- 
sion, or  why  "more  gives  less,  and  less  more."  Nor 
have  any  of  them  so  reasoned  on  the  assumption,  as 
to  give  the  pupil  even  their  own  vague  and  indefinite 
ideas ;  much  less  satisfying  his  common  sense  with  a 
rational  and  demonstrable  theory.  No  doubt  that  the 
best  arrangements  practicable  have  been  made ;  for  it 
is  wholly  impossible  to  explain  the  theory  on  any  other 
principles  than  those  of  cause  and  effect.  These  are 
the  very  embodiments  of  inverse  action ;  the  one  go- 
ing directly  on  to  create ;  the  other  returning  to  the 
creator  or  cause,  from  the  created  effect.  The  obe- 
dient pupil  has  too  long  followed  in  the  wake  of  rules 
that  linger  along  in  obscurity  and  darkness,  not  only 
in  this  but  in  most  of  the  departments  of  this  beauti- 
ful science. 

How  many  yards  of  cloth,  4  quarters  wide,  are 
equal  to  40  yards,  5  quarters  wide  ? 


5 


Here,  4  quarters  is  the  demand,  5  quarters 
the  same  name,  and  40  yards  the  term  of  an- 
swer. 1 50 

If  it  take  20  yards  of  cloth,  f  of  a  yard  wide,  to 
make  a  gown,  how  many  yards,  £  wide,  will  it  take  to 
line  it? 

The  demand,  £,  is  placed  on  the  left,  by  I  7  $     2 
its  numerator;  while  the  same  name,  f,  is,  in 
the  same  manner,  placed  on  the  right. 


17f 

How  many  yards  of  silk  serge,  2£  yards  wide,  will 


160 


RAINEY'S  IMPROVED  ABACUS. 


line  a  coat,  that  requires  15  yards  of  cloth,  f  of  a 
yard  wide  ? 


Fifteen  yards  being  the  denomination 
of  the  answer,  is  placed  last  on  the  right. 


5  yds. 


If  5  yards  long,  and  2£  yards  wide,  line  a  coat,  how 
long  must  the  piece  of  cloth  be,  which  is  f  of  a  yard 
wide,  to  make  it? 

t* 

ft  p     o  rpjjg  demand  is  %  in  this  case,  while  the 

_  5 same  name  is  2|-.     Again: 

15yds. 

If  15  yards  long  and  f  wide  make  a  coat,  how  wide 
must  5  yards  of  serge  be,  to  line  it  ? 

In  this  case,  the  two  terms  to  be  com- 
pared, are  in  length ;  while  width  is  the  de- 
nomination of  answer.  The  serge  must  be 
2^  yards  wide  ? 

If  5  yards  long  and  2^  wide,  make  a  coat,  how  wide 
must  15  yards  in  length  be,  to  line  it  ? 


m-i 

43 


t-tt 

4 

£ 
0-3 

4 

3 

4 

The  answer  is  f  of  a  yard  wide. 


It  is  perceived,  that  these  questions  are  inverse,  be- 
cause all  of  the  terms  are  causes  of  capacity. 

Suppose  a  garrison  has  provisions  to  last  4  months, 
at  20  ounces  per  day ;  how  many  ounces  should  they 
use,  for  it  to  last  6  months  ? 


INVERSE    PROPORTION. 


161 


The  two  different  number  of  months     3 — 014 
must  here  be  compared ;  6  months  being  mQ 

the  demand,  and  20  ounces  the  term  of          "lilTi 
answer.  I     » 

If,  when  wheat  is  worth  60  cents  per  bushel,  the 
cent  loaf  weighs  10  ounces,  how  much  ought  it  to 
weigh  when  wheat  is  worth  $1,25  per  bushel? 


The  prices  are  compared,  as  causes,  and 
the  weight  is  the  term  of  answer. 


125 


60 
10 

4* 


If,  when  wheat  is  worth  $1,25  cents,  the  cent  loaf 
weighs  4f  ounces,  what  ought  it  to  weigh  when  wheat 
is  60  cents  per  bushel  ? 

60  125 


The  answer  is  10  ounces. 


24 


10  oz. 

If  a  bushel  of  wheat  make  40,  five  cent  loaves, 
how  many  eight  cent  loaves  will  it  make  ? 

The  ratio  is  obtained  between  the  8  and 
5  cents,  as  the  two  most  active  causes  con- 
ducing to  the  effect.  Thus,  capital  is  at 
times  a  cause. 


25 


If  it  require  110  yards  of  carpeting  f  of  a  yard 
wide,  to  carpet  a  room,  how  many  yards,  1-f  yards 
wide,  will  carpet  the  same? 

tfaf— 2 


The  answer  is  60  yards. 


60 


If  I  lend  a  friend  $100,  for  40  days,  how  long  ought 
he  to  lend  me  $80,  to  return  the  accommodation? 

>  100—5 
The  answer  must  be  in  days. 


"J50  da. 


162  RAINEY'S  IMPROVED  ABACUS. 

How  long  must  a  board  be,  that  is  9  inches  wide, 
to  make  a  square  foot? 

The  question  should  be, 

If  12  inches  long  and  12  inches  wide,  make  1  foot, 
what  length,  with  9  inches  in  width,  will  make  the 
same? 

Nine,  the  width,  is  evidently  the  demand,  while  12  in 
width  is  the  same  name,  and  12  in  length  is  the  term 
of  answer;  for  the  answer  is  required  in  length. 

— $  4-4 — 4         Without  transferring  and  working  the 
jLA — 4     question   anew,  we   may  prove   it  by 
simply  multiplying   9    and   16,   which 
make  144,  the  number  of  square  inches 
in  a  square  foot. 

If  a  certain  pasture  supply  1500  cows  75  days,  how 
long  will  it  supply  2000  cows  ? 


16  in. 


2000 


1500 

75 


The  causes  are,  in  this  instance,  active 
causes. 

If  a  man  perform  a  journey  in  24  days,  when  the 
days  are  12  hours  long,  how  many  days  will  he  be  re- 
quired to  do  the  same,  when  the  days  are  18  hours 
long? 

The  causes  here,  are  all  of  the  same 
kind,  time ;  yet  we  know  that  the  an- 
swer must  be  in  days,  and  place  the 
days,  consequently,  on  the  right. 
We  will  now  present  a  few  questions  in  the  calcula- 
tion of  machinery ;  all  of  which,  so  far  as  velocity  is 
concerned,  are  inverse. 

If  a  drum,  60  inches  in  diameter,  make  72  revolu- 
tions per  minute,  how  many  revolutions  will  a  pulley 
connected  with  it,  make,  which  is  6  inches  in  diameter  ? 
It  is  necessary,  in  this  question,  to  compare  the  two 
diameters ;  and,  as  6  diameter  is  the  demand,  we  place 


CALCULATION    OF    MACHINERY. 


163 


it  on  the  left,  and  60  diameter  on  the  right ;  while  72, 
the  number  of  revolutions,  becomes  the  denomination 
of  the  answer. 

The  answer  is  720  revolutions  _ 

720 

Let  us  now  change  this  question,  and  find  the  diam- 
eter of  this  pinion,  knowing  the  number  of  revolu- 
tions that  we  wish  to  make. 

If  72  revolutions  require  60  inches  in  diameter,  of 
what  diameter  must  a  pinion  foe,  to  make  720  revo- 
lutions ? 

NAM 

Thus,  the  diameter  used  at  the  outset  60 

is  again  found.  "~  /T~I 

Again:  Making  the  two  diameters  of  the  demand 
thus  found,  the  supposition,  we  will  endeavor  to  find 
#ach  term  of  the  former  supposition,  72  and  60. 

If  6  inches  in  diameter  nmke  720  revolutions,  how 
many  revolutions  will  60  inches  diameter  make  ? 


Here,  the  diameters  must  be  compared. 


720 


72  rev. 

If  720  revolutions  require  6  inches  diameter,  what 
diameter  will  give  72  revolutions  ? 

Thus,  we  can  find  the  diameter  and 
revolution  of  any  wheel,  the  diameter  and 
revolutions  of  that  with  which  it  is  con- 
nected, being  given. 


6 


60  in. 


It  frequently  becomes  necessary  to  Ascertain  the 
revolutions  or  -diameter  of  a  wheel  connected  with 
several  others.  This  is  done  by  what  is  called  "  con- 
joined proportion,"  which  our  limited  'space  will  not 
permit  us  to  treat  here- 


10  in. 


200  in. 


164  RAINEY'S  IMPROVED  ABACUS. 

Suppose  a  counter  wheel  in  a  mill  turn  40  times  per 
minute ;  how  large  must  a  trundle  be  on  a  spindle,  to 
make  240  revolutions  per  minute,  the  counter  wheel 
being  5  feet  in  diameter  ? 

Two  hundred  and  forty  revolutions  is 
the  demand,  40  revolutions  the  same 
name,  and  5  feet,  or  60  inches,  the  diam- 
eter of  the  counter  wheel,  is  the  term  of 
answer.  Hence,  the  trundle  must  be  10  inches  in 
diameter. 

How  many  inches  in  diameter  must  a  water-wheel 
be,  to  make  12  revolutions,  if  a  counter  wheel,  60 
inches  in  diameter,  makes  40  revolutions  per  minute? 

£&4Q  These    200    inches    make   16|  feet, 

which  might  be  easily  obtained  by  substi- 
tuting 5  feet,  for  60  inches,  when  the  an- 
swer would  be  in  feet.  This  shows  us, 
that  a  water-wheel  16-j-  feet  in  diameter,  making  12 
revolutions  per  minute,  working  in  a  crown  wheel  of 
equal  size,  with  a  counter  wheel,  60  inches  in  diame- 
ter, making  40  revolutions,  must  have  a  trundle  10 
inches  in  diameter,  to  make  240  revolutions  per 
minute.  We  may  go  on  thus  from  one  wheel  to  an- 
other, to  any  extent,  and  apply  the  diameter  and  rev- 
olutions of  one  to  another,  coming  in  regular  sequence 
after  it,  until  we  ascertain  the  revolutions  and  diame- 
ters in  a  long  chain  of  machinery. 

Such  calculations  as  the  following,  are  frequently 
necessary  in  finding  the  size  of  wheels,  their  "  pitch- 
line,"  and  the  number  of  "  cogs,"  or  teeth.  They 
come  properly  under  the  head  of  direct  proportion ; 
but  are  placed  in  this  connection,  that  most  of  the  re.- 
marks  of  machinery  may  be  found  together.  The 
problems  are  solved  as  other  questions  in  simple 
proportion. 

Suppose  it  be  required  to  make  a  wheel  1^  inches 
larger,  whose  pitch-line  ia  12 if  inches  in  diameter  t 


INVERSE  PROPORTION:   MACHINERY.         165 

how  many  cogs  will  the  wheel  so  enlarged  have,  if  the 
first  wheel  had  67  cogs  ? 

We  first  add  1%  to  12}f  inches,  making  thus,  12-|- 
1=13  inches :  ^  inch  equals  T8F,  and  8-J-15  sixteenths 
equal  f-f ,  or  177F,  which  added  to  13  makes  14y7F,  the 
diameter  of  the  pitch -line,  after  adding  the  1^  inches. 
We  now  say,  as  12Tf  is  to  14T7g-,  so  will  67  cogs  be 
to  the  number  required. 

If  a  wheel  12-J-J-  inches  in  diameter,  have  67  cogs, 
how  many  cogs  will  a  wheel  14T7g-  inches  in  diameter 
have;  thus, 


231 

16 

67 


The  demand  is  here  placed  on  the  right.  16 
The  wheel,  thus  increased  in  diameter,  will  207 
have  74|f  cogs;  and,  as  the  number  of 
cogs  must  always  be  even,  we  will  sup- 
pose  that  the  wheel  shall  have  75  cogs, 
being  the  whole  number  that  nearest  expresses  the 
fraction.  Now,  this  will  require  the  pitch-line  to  be 
still  a  little  larger.  So  we  say,  if  67  cogs  be  ad- 
vanced to  75  cogs,  what  will  the  pitch-line,  12Tf ,  be 
advanced  to  ?  Seventy-five  cogs  is  the  demand,  67 
cogs,  the  same  name,  and  12}£  in  diameter,  the  term 
of  answer. 


67 
16 


75 
207 


The  answer  is  a  very  minute  fraction 
under  14^-  inches;  and  this  pitch-line 
can  be  made  so  nearly  the  right  diame- 
ter, as  to  step  off  with  the  "  dividers  " 
75  cogs,  without  any  perceivable  fraction. 

Suppose,  again,  a  wheel,  that  is  18  inches  in  diam- 
eter, has  100  cogs;  how  many  cogs  must  it  have  to 
be  enlarged  6  inches  ? 

Here,  18-|-6=24,  the  diameter  of  the  required 
wheel.  Now,  how  many  cogs  will  24  the  desired  di- 
ameter require,  if  18  inches,  the  given  diameter,  re- 
quire 100  cogs;  thus, 


166  RAINEY'S  IMPROVED  ABACUS. 


18 


24 

100 

1334- 


This  gives  for  the  number  of  cogs  in  the 
larger  wheel  133-J,  which  not  being  an 
even  number,  we  call  133,  that  from  this 


smaller  number  of  cogs,  we  may  make  the 
new  pitch-line  proportionally  smaller. 

If  100  cogs  require  18  inches  diameter,  what  pitch, 
diameter,  will  133  cogs  require? 


100 


133 

18 


I"** 


Therefore,  to  make  133  cogs,  which  is 
an  even  number,  the  new  pitch-line  must 
be  23|J,  or  23.94  decimal  inches;  which, 


too,  can  be  so  accurately  approximated,  as 
to  give  133,  in  about  the  desired  diameter. 

The  article  on  Inverse  Proportion,  although  short, 
has,  we  trust,  a  sufficient  number  and  variety  of  ex- 
amples to  enable  the  judicious  and  discriminating 
reader  to  apply  the  principles  in  practice  whenever 
found  necessary. 

We  now  close  our  remarks  on  the  proportions;  and 
shall  hereafter  advert  to  them  only  to  show  the  philo- 
sophy of  the  statements  that  follow  in  the  various  rules 
of  arithmetic  and  mensuration :  as  all  of  these  subdivi- 
sions depend  in  principle  and  detail  on  the  proportions ; 
and  are  but  different  branches  of  them,  assuming  such 
apparently  different  form,  as  the  circumstances  of  their 
application  may  require.  We  feel  confident,  that  if 
the  reader  will  carefully  review  the  foregoing  pages, 
with  a  close  attention  to  principles,  rather  than  prac- 
tice, he  will  find  no  difficulty  of  perceiving  the  neces- 
sary operations  to  be  performed  on  all  questions  that 
will  follow  in  this  work;  nor  in  applying  the  principles 
that  he  has  learned,  in  such  manner  as  to  conduct  to 
satisfactory  results.  He  cannot  expect  to  do  this,  by 
a  slight  and  indifferent  course  of  reading :  progress  in 
arithmetical  knowledge  justifies  no  such  conclusion,  if 
we  consult  the  history  of  those  who  have  attained 
great  proficiency  in  its  principles.  To  become  tho- 
roughly acquainted  with  the  science,  requires  not  only 


INVERSE  PROPORTION:  RULE.  167 

a  due  comprehension  of  each  separate  principle,  but  a 
knowledge  of  all  their  relative  bearings;  and  even 
with  these,,  such  an  acquaintance  and  familiarity,  as 
to  render  their  constant  contemplation,  which  ever 
enables  us  to  discern  new  beauties,  rather  a  pleasure 
than  a  task.  Nor  must  we  regard  cancelation  as  a 
short-hand  system,  merely,  that  will  enable  us  to  ar- 
rive at  results  instanter,  without  that  thought  neces- 
sary to  the  apprehension  of  great  general  principles  in 
any  science :  for  the  abbreviation  of  this  system,  con- 
sists, not  in  the  shortening  of  principles,  but  in  their 
use  and  concentration.  If,  moreover,  we  investigate 
a  proposition  for  the  beauty  and  grandeur  of  its  prin- 
ciples and  relations,  instead  of  as  a  forced  and  neces- 
sary task,  for  merely  speculative  purposes,  we  will 
not  fail  being  thoroughly  acquainted  with  it;  thus 
connecting  the  pleasures  of  the  science,  with  the  neces- 
sity of  the  art. 

From  the  foregoing  illustrations,  we   deduce  the 
following 


SUMMARY   OF    DIRECTIONS    FOR   INVERSE   PROPORTION. 

Ascertain  first  the  term  of  Demand : 
Place  this  Demand  on  the  left : 

Place  the  term  of  the  same  name  of  the  demand, 
opposite  the  demand,  on  the  right: 

Place  the  term  in  ichich  the  answer  is  required,  last 
on  the  right. 

If  any  of  the  terms  are  fractional. 

Place  the  Numerators  on  the  side  of  the  line  ordina- 
rily assigned  to  the  integers,  and  the  Denominator 
opposite 


168  RATNEY'S  IMPROVED  ABACUS. 


CONJOINED  PKOPORTION. 

Conjoined  Proportion  differs  from  Simple  Propor- 
tion, only  by  reason  that,  instead  of  one,  many  differ- 
ent proportions  are  found  in  one  question.  Hence, 
all  questions  coming  under  this  head,  may  be  wrought 
by  continued  separate  statements  in  Simple  Propor- 
tion. The  demand  is  placed  on  the  right,  and  the 
same  name,  in  the  supposition,  on  the  left.  This  sup- 
position merely  equals  in  value  the  term  in  which  the 
answer  is  required,  and  which  is  placed  invariably  on 
the  right.  Hence,  statements  in  this  species  of  pro- 
portion are  always  direct.  Now,  the  term  which  is  of 
the  same  name  of  that  just  used  as  denomination  of 
answer,  may  be  placed  opposite  this  denomination  of 
answer,  and  the  new  denomination  of  answer  which 
this  second  term  of  supposition,  or  same  name,  equals, 
may  be  placed  last  on  the  right  for  a  new  term  of  an- 
swer. So,  again,  this  term  of  answer  may  be  used  as 
demand,  with  another  supposition  opposite,  and  an- 
other term  of  answer  succeeding,  until  any  number  of 
separate  statements  are  merged  into  one  general  state- 
ment. Hence,  the  statement  will  be  but  a  concatena- 
tion or  chain  of  statements ;  from  this  fact,  it  has  re- 
ceived the  name  of  " chain  rule"  Nothing  more  is 
necessary,  to  make  this  interlinking  of  statements 
plain  and  easy,  than  to  find  the  term  of  demand,  or 
that  term  for  which  an  equivalent  term  in  value,  is 
demanded.  It  will,  therefore,  be  necessary,  to  place 
terms  of  the  same  denomination,  and  such  only,  oppo- 
site one  another. 

If  5  Ibs.  of  raisins  are  worth  81bs.  of  tamarinds, 
and  3  Ibs.  of  tamarinds  are  worth  18  Ibs.  of  figs,  and 
20  Ibs.  of  figs  are  worth  75  Ibs.  of  cheese,  and  40  Ibs . 


CONJOINED    PROPORTION. 


169 


of  cheese  are  worth  16  Ibs.  of  butter,  and  l&f  Ibs. 
butter  are  worth  30  yards  of  calico,  and  5  yds.  calico 
are  worth  4J-  bushels  of  apples,  how  many  bushels  of 
apples  are  worth  13^  Ibs.  of  raisins  ? 

It  is  evident,  that  the  answer  must  be  in  bushels  of 
apples,  and  that  the  demand  is  13^  Ibs.  of  raisins: 
hence,  we  conclude,  that  if  raisins  is  the  demand, 
raisins  must  be  the  same  name;  and  consequently, 
place  opposite  the  13f  Ibs.  raisins,  the  term  of  raisins 
given  in  the  question,  which  is  5  ;  and  so  on  with  all 
of  the  other  terms,  making  the  8  Ibs.  tamarinds,  which 
this  5  Ibs.  equals,  the  term  of  answer,  under  the  de- 
mand, and,  again,  the  3  Ibs.  of  tamarinds,  the  same 
name,  opposite,  and  so  on  ;  thus, 


This  may  be  proven  by  changing 
the  conditions  of  the  question. 
Again : 


t 


25 


30 


7128 


If  3  days  work  of  A,  are  equal  to  5  of  B,  and  4 
of  B  to  9  of  C,  and  10  of  C  to  6  of  D,  and  8  of 
D  to  40  of  E,  and  3  of  E  to  2£  of  F,  and  3|  of  F 
to  18f  of  G,  how  many  days' work  of  A  will  equal 
25  of  G? 

Here,  G-  is  the  demand,  and  certainly  G  must  be 
the  same  name ;  hence,  we  state  in  the  retrograde 
order  of  the  question,  until  we  find  A,  which  was  the 
first  term  proposed,  and  which  is  now  the  term  of  an- 
swer; thus, 


170 


RA1NEY  S    IMPROVED    ABACUS. 


3- 


15 


15 


of  A. 


The  result  is  T8-j  of  1  day's  work 
of  A.  The  proportion  in  these 
questions  is  always  direct ;  because 
the  number  on  the  left  merely  equals 
in  value  some  other  member  on  tLe 
right.  Again : 


If  $18  U.  S.  are  worth  8  ducats,  at  Frankfort;  12 
ducats  at  Frankfort,  9  pistoles,  at  Geneva;  50  pistoles 
at  Geneva,  40  rupees,  at  Bombay;  16  rupees  at  Bom- 
bay, 20  rials,  at  Buenos  Ayres ;  15  rials  at  Buenos 
Ayres,  6  candarines,  at  Canton;  4J-  candarines  at 
Canton,  8  lepta,  at  Corinth ;  2  lepta  at  Corinth,  5  as- 
pers  at  Cairo ;  and  20  aspers  at  Cairo,  40  copecks,  at 
St.  Petersburgh;  how  many  copecks  of  St.  Peters- 
burg are  worth  $120,  U.  States  ? 


The  answer  is  142f  co 
pecks. 


Dol.  18 

120  Dollars. 

Due.  12 

8       Ducats. 

Pis.   50 

9      Pistoles. 

Bup.  16 
Ri.    15 

40     Rupees. 
20     Rials. 

Can.    9 

6       Candarines. 

2 

Lep.    2 

As.    20 

8       Lepta. 
5       Aspers. 
40     Copecks. 

142f 

All  questions  of  a  similar  nature,  in  exchange  and 
reduction  of  monies,  may  be  wrought  as  the  above, 
and  without  the  least  difficulty,  when  proper  attention 


EQUATION    OF    PAYMENTS.  171 

is  given  to  the  foregoing,  which  may  be  brought  into 
the  following 

SUMMARY   OP    DIRECTIONS    FOR   CONJOINED   PROPORTION. 

Place  the  demand  first  on  the  right:  Place  the 
same  name  opposite ;  and  place  the  term  of  answer, 
which  equals  the  same  name,  under  the  demand,  on  the 
right.  Again:  Place  opposite  the  term  of  answer, 
the  term  of  the  same  name,  and  the  term  which  equals 
the  same  name,  under  the  former  term  of  answer, 
on  the  right ;  and  so  on,  until  all  of  the  terms  are 
placed  down.  Cancel  as  in  other  cases. 


EQUATION  OF  PAYMENTS. 

It  frequently  becomes  necessary  to  make  several 
different  payments  on  a  note  at  one  time,  or  to  make 
one  payment  of  several  different  notes.  This  is  done 
by  liquation  of  payments,  which  simply  means,  to 
make  the  time  of  one  payment  equal  to  the  average 
of  the  several  periods.  Equation  is  from  the  Latin, 
equa,  equal.  The  following  facts  may  be  assumed : 


1,  in    3  months,  will  gain  as  much  as  { 

510,  in   8       "  "  "  "  "  < 

>40,  in    1       "  "  "  "  " 

4,  in  20  days  "  "  "  "  l 

S17,  in  11  "  "  "  "  "  ' 

$,  in    1  year  "  "  "  "  ' 

*19,  in    7      "  "  "  "  " 


3  in  1  mo 
8  in  10  « 
1  in  40  " 

^20  in    4  da. 

;ll  in  17    " 
1  in    8yrs. 
7  in  19    " 

A  merchant  owes  two  notes,  payable  as  follows :  one 
for  $10,  payable  in  8  months,  and  the  other  for  f 
payable  in  6  months.     Now, 


172 


RA1NEY  3    IMPROVED    ABACUS. 


8  months  equals  $  80,  for  1  month. 
40X6      "  "      $240,  for  1      " 

50||320,  for  1      « 
64  months. 


$50 


Above,  each  sum  is  multiplied  by  its  number  of 
months.  The  original  sums  are  added,  making  $50, 
for  the  equated  time ;  or,  their  products  in  the  months, 
making  $320,  for  1  month.  The  latter  product  is 
divided  by  the  former,  giving  the  equated  time  6f 
months.  The  question  would  be  stated  thus,  by  In- 
verse Proportion : 

If  $320  shall  be  paid  in  1  month,  in  how  many 
months  must  $50  be  paid  ?  Thus, 

50  320  This  is  the  proper  method  of  arriving  at 

1  the  proportion  in  such  questions.     The  an- 

~~  7JT"  swer  is,  as   before,  6f    months ;  because  1 

1  J  '  month  was  the  last  term  on  the  right. 

Again :  A  owes  to  B  4  notes,  payable  as  follows : 


;  25  X  in    6  months  =  9  150  ] 
,  75x  in    8       "       =$  600   ( 
200  X  in  10       "       =$2000   f1 
i  60X  in    3       "       =  $  180  J 


1  month. 


$2930 


360 


2,930 


Ans.  8/g-,  equal  to  8  months,  4 
days,  and  4  hours. 

I  give  2  notes,  one  for  $100,  payable  in  6  months; 
the  other  for  $100,  payable  in  18  months:  at  what 
time  may  both  be  paid  together  ? 


RULE  FOR  EdUATION  OF  PAYMENTS.    173 

$100 X-  6  months  =  $  600  )  ,     1         ., 
j$100xl8       <£       =  $1800  (  h* 

$200  #00  #00—12 


12  months,  answer. 

This  method  of  calculating  equation  of  payments, 
is  not  strictly  correct ;  being  based  on  the  supposition 
that  interest  and  discount  are  the  same ;  that  is,  that 
the  deduction  made  in  advance,  is  equal  to  the  interest 
that  accrues,  after  a  certain  period.  Thus,  in  the 
question  above,  $100  are  withheld  from  the  creditor  6 
months,  and  interest  should  be  charged ;  whereas,  an- 
other $100  were  paid  6  months  before  due,  and  should 
sustain  only  a  discount.  The  interest  on  the 
money  withheld,  is  greater  than  the  discount  on 
that  advanced;  hence  the  difference.  This  differ- 
ence is,  however,  very  minute,  and  of  no  practical 
importance. 

When  the  time  is  days  or  years,  such  days  or  years 
must  be  multiplied  into  the  sum  to  be  paid,  as  in  the 
cases  of  months  above ;  and  the  equated  time  will  be 
days  or  years,  as  the  case  may  be.  When  the  time  is 
months  and  days,  it  must  be  reduced  either  to 
months  or  days. 

If  one  of  the  debts  is  paid  down,  or  a  payment 
made  at  the  date  of  the  obligation,  it  will  make  no 
product  in  multiplying  into  time,  and  will  be  used  only 
in  finding  the  sum  of  the  debts  or  payments. 

From  the  foregoing,  we  deduce  the  following 

DIRECTIONS    FOR    EQUATION    OF    PAYMENTS. 

Multiply  the   sum  of  each  payment  by  its  time, 
add  the  several  payments:  then,  add  their  products ; 
and  divide  the  sum  of  the  products,  by  the  -sum  of  the 
payments. 
12 


174  RAlNEYyS   IMPROVED    ABACUS  r 

The  several  periods  of  time  must  be  of  the  same 
denomination  ;  either  days,  months,  or  ytars,  separate- 
ly ;  and  the  answer  will  be  the  equated  time,  in  days, 
months,  or  years,  as  the  ease  may  be.  Or,  state  as  m 
Inverse  Proportion. 


FELLOWSHIP,  SIMPLE  AND  COMPOUND 

INCLUDING  PARTNERSHIP,  GENERAL  AVERAGE,  BANK- 
RUPTCIES, ETC, 

In  Simple  Fellowship  or  Partnership,  General  Ave- 
rage, Bankruptcy,  etc.,  two  or  more  individuals,  on- 
different  sums  of  money,  gain  or  lose  some  general  sum ; 
when,  it  is  desired  to  know  each  individual  gain  oir 

loss. 

In  Compound  Fellowship,  not  only  the  sums  of  cap- 
ital are  different,  but  are  invested  for  different  peri- 
ods of  time,  etc.  From  this  fact  it  takes  the  name  of 
Compound  Fellowship. 

A,  B,  and  C  invest  $1000  in  a  cargo  of  wheat : 
A  puts  in  $200,  B  $300,  and  C  $500 ;  and  agree  to 
share  the  profits  and  losses,  in  proportion  to  the  capi- 
tal severally  invested.  They  gain  $600 ;  what  is  each 
man's  share? 

A's  share  $200 
B's     "       300 

C's     "       500 

$1000.     Sum  of  gain  $600. 

Now,  fhe  whole  sum,  1000,  gains  the  whole  sum, 
600 ;  therefore,  we  may  ask,  what  share  of  600  each 


SIMPLE    FELLOWSHIP. 


175 


individual  share  gains.     The  statements,  according  to 
proportion,  are  as  follows : 


If  1000  gain  600,  what  will  200  gain? 
«      «       «       u        a       «    300     " 
«      «       «       «        «      «    500     " 


£000200 


$120 


300 
600 


500 
600 

300 


A's  share  of  gain  120 
B's  "  "  "  180 
O's  "  '•  "  300 


Whole  sum  gained,  $600  Proof. 

A  father  divides  $1800  among  three  sons,  in  the 
following  proportion :  A,  1  share ;  B,  twice  as  much 
as  A ;  and  C,  three  times  as  much  as  B  ;  what  is  the 
share  of  each? 

Let  us  set  A's  share  down  as  a  unit ;  thus,         1 
B's  share,  as  twice  this,  2 

C's  share,  as  three  times  the  latter,  6 

Sum  of  the  shares,  9 

Thus,  there  are  9  shares ;  and  A  has  £ ;  B,  f ,  and 
C,  £ :  hence,  their  shares  would  be  as  9  to  1 ;  9  to  2 ; 
and  9  to  6 ;  thus, 

0^00—2  #,i#00— - 2 

2 


200 


400 


6 


1200 


Or,  the  questions  might  be  stated,  thus, 
What  will  1  share  be,  if  9  shares  be  1800?  What 
will  2  shares  be  ?     What  wiU  6  shares  be,  etc.?     200 
_j_400+1200=$1800,  the  estate ;  which  proves  the 
work  correct. 


176 


RAINEY  S   IMPROVED    ABACUS. 


A  owes  to  B  $400; 
"  "  "  C  $600; 
"  "  "  D  $500; 
"  "  "  E  $800; 
"  "  "  F  $200 ;  and  pays  only 
much  does  each 


$2500 
R600:  how 


lose? 


$2500 


$900 


The  whole  sum  of  the  credits  loses  $900 :  hence,  the 
following  statements: 


££00 

400              ££00 
900—4 

600          ?£00 
900  —  4 

500 
900—4 

$ 

144  B 

££00 

g 

V 

800 
900—4 

216  C. 

n<M> 

to 

¥ 

200 
900- 

180  D. 
1 

144+216+180+288+72=900  dollars,  the  whole 
Bum  lost. 

A  man  left  to  his  four  sons  $60,  to  be  divided  in 


the   proportion  of  -J-, 

one  get  ? 

i  of  60=20 
|  «  60=15 
i  "  60=12 
i-  «  60=10 


£,  i;  how  much  does  each 


57 


20 
60 


is  57  : 

60 

::  20  : 

X  or  21/T 

"  57  : 

60 

:  :  15  :  X  or  15^5. 

"  57  : 

60 

::  12  :  Xorl2-f4 

"  57  : 

60 

::  10  : 

XorlOfl 

$60 

5715 

57 

12 

57 

10 

60 

8 

60 

60 

'l  ZA. 

1On  n 

1H3  0 

I4-**?  [Avi7 

The  several  sums,  added,  make  the  original  $60. 


GENERAL    AVERAGE    IN    FELLOWSHIP.  177 

The  $57  was  the  deficient  sum  of  capital;  and  20, 
15, 12,  and  10,  the  deficient  individual  sums.  Then,  if 
57,  deficient,  be  advanced  to  60,  full  sum,  what  will 
any  of  the  other  deficient  sums  be  advanced  to,  for 
full  sum  or  share  ? 

Railroad,  canal,  bank,  and  other  stock  dividends,  or 
assessments,  may  be  calculated  as  the  first  and  second 
examples  given. 

GENERAL  AVERAGE. 

It  frequently  becomes  necessary,  at  sea,  to  throw 
overboard  a  large  portion  of  the  cargo,  to  secure  safety 
in  time  of  storm.  The  property  thus  sacrificed  may 
belong  to  one  individual,  although  it  is  thus  thrown 
overboard  for  the  benefit  of  the  whole.  Hence,  the 
whole  cargo  should  sustain  a  loss  proportionate  to  the 
value  of  each  individual  interest.  Now,  it  becomes 
necessary  to  assess  the  loss  according  to  value :  this 
is  called  General  Average ;  while  the  property  ejected 
is  called  jettison,  from  the  French,  jetter,  to  throw. 
All  such  losses  are  sustained  by  the  ship,  the  cargo, 
and  the  freight,  according  to  the  value  of  each,  which 
is  called  pro  rata,  or  in  proportion. 

Ordinary  losses,  such  as  wear  and  damage,  or  sacri- 
fice made  for  the  safety  of  the  ship  alone,  must  be 
borne  by  the  owners  of  the  vessel;  losses  made  for 
the  safety  of  any  particular  portion  of  the  cargo,  must 
be  charged  to  the  individual  so  losing,andarenot  to  be 
brought  into  the  general  average. 

The  property  lost  must  be  reckoned,  as  well  as  that 
saved. 

The  cargo  is  valued  at  the  price  it  would  bring  at 
the  place  of  destination,  after  deducting  storage  and 
all  necessary  charges. 

One-third  is  generally  deducted  from  the  freight,  for 


178  RAINEY'S  IMPROVED  ABACUS. 

wages,  pilotage,  etc.,  etc.;  in  New  York,  one-half  is 
allowed. 

One- third  of  the  cost  of  repairs,  on  masts,  spars, 
rigging,  etc.,  is  deducted  before  the  average  is  made; 
thus  making  the  valuation  of  the  old.  two-thirds  of 
the  new.  This  is  done  on  the  principle  of  insuring 
only  two-thirds  of  the  value  of  property  on  land;  or  to 
allow  for  damage.  And  here,  the  danger  must  be 
imminent,  or  the  general  average  will  not  be  allowed. 

All  necessary  charges  must  be  deducted  from  each 
individual's  interest  before  the  average  is  made. 
Example : 

The  ship,  John  Adams,  from  Havre  to  Boston,  had 
on  board  a  cargo  estimated  at  $60,000.  Of  this  A 
owned  $20,000;  B,  $30,000;  and  C,  $10,000.  The 

Eoss   amount    of    freight   and   passage    money   was 
.2,000.     The  ship  was  worth  $50,000;  and  $800 
d  been  paid  for  insurance.     The  ship,  being  in  great 
distress,    the  master  threw  overboard  $7,800  worth 
of  goods,  and  cut  away  her  masts,  rigging,  and  an- 
chors.    In  port,  it  cost  $3,000  for  repairs;  what  was 
the  loss  of  each  owner,  both  of  ship  and  cargo. 

Ship  valued  at      ....     $50,000 

Premium  deducted,    .     .     .  800         $49,200 

Cargo  worth,        '  60,000 

Freight  and  passage,      .     .     .12,000 

One-third  deducted  for  wages,     4,000  8,000 

Sum  of  individual  interests,    .     .     .     .    $117,200 

Goods  thrown  overboard, 7,800 

Cost  of  new  masts,  spars,  etc.,  $3,000 

One-third  deducted  for  wear,       1,000  2,000 

Com.  on  repairs, 20 

Port  duties  and  other  expenses,    ...  180 

Sum  of  loss,    ...  .     .     $10,000 


•COMPOUND    FELLOWSHIP. 


179 


As  117,200  :  20,000  :: 
As       "       :  30,000  :: 
As       "       :  10,000  :: 
As       «        :  49,200  :: 
As       «       :     8,000:: 

PROOF.    Whole  loss, 

10,000  .:  $1706.48  A's  loss 
«       :    2559.73  B's   « 
«       :       853,24  C's    " 
«       :    4197.95  Ship's  1. 
«       :       682.60  Frt's.  L 

$10,000.00  as  ab've. 

COMPOUND   FELLOWSHIP. 

A,  B,  and  0  purchase  a  pasture  to  be  used  by  them 
jointly,  for  $50,  in  which  A  keeps  80  oxen  3  months; 
B,  100  oxen  2  months;  and  C,  160  oxen  1  month: 
what  part  of  the  cost  must  each  pay  ? 

It  is  not  necessary  here  that  the  time  be  expressed 
.as  months ;  for  it  is  simply  a  unit ;  and  3,  2,  and  1 
show  the  ratios  of  consumption  of  grass,  rather  thaa 
the  time^  as  the  $50  pay  alike  for  1  month,  1  year, 
or  10  years.  The  multiplication  of  the  time  and  oxea 
together,  shows  -merely  how  many  oxen,  in  each  case, 
remain  a  common  length  of  time  in  the  pasture ;  for, 
after  being  thus  multiplied,  one  lot  is  supposed  to  be 
in  the  pasture  as  long  as  the  other;  and  the  difference 
of  consumption  and  price  is  the  effect  of  the  different 
numbers  of  oxen  thus  grazing. 

A's    80x^  months  -=  240  oxen  for  the  time. 
B's  100x2      "        =  200    "       "     "      « 
C's  160x1      "       •=  160    "       "     «      " 

The  whole  =  600    «       "    •«      -" 

Now,  if  600  oxen  cost  $50,  what  will  240, 200,  and 
160  cost,  respectively;  thus, 


600 


240 
50 


600 


20  A's. 
PJEIOOF. 


200 
50 


eoo 


16|  B's. 


160 
50 


180 


RAINEY'S  IMPROVED  ABACUS. 


A  and  B  companied;  A  put  in  $2,000,  January 
1 ;  but  B  put  in  June  1 ;  what  sum  did  he  put  in,  to 
have  an  equal  share  of  the  profits  with  A  ? 

A  has  2,000  employed  12  months ;  it  is,  therefore, 
desired  to  know  how  much  capital  belonging  to  B  will, 
in  7  months,  make  equal  profits  with  the  2,000,  12 
months.  Hence,  the  question  is  one  of  inverse  pro-- 
portion,  and  is  wrought  thus, 


12 

2000 


The  demand  is  on  the  left. 


$  3428A 

" In  an  adventure,  A  put  in  $12,000  for  4  months; 
then  adding  $8,000,  he  continued  the  whole  2  months ; 
B  put  in  $25,000,  and  after  3  months  took  out  $10,000, 
and  continued  the  rest  3  months  longer;  C  put  m 
$35,000  for  2  months ;  then,  withdrawing  f  of  his 
stock,  continued  the  remainder  4  months  longer :  they 
gained  $6,000 ;  what  was  the  share  of  each  ? 


=  88,000. 


A'sj 
A's! 

$12,000x4  BIOS. 
§20,000x2     " 

=    48,000, 
=    40,000, 

B'sj 
B's< 

$25,000X3     " 
|15,OOOX3     " 

=    75,000, 
=    45,000, 

C's  $35,000x2     "• 
C's  $25,000x4     " 

Whole  sum, 

=    70,000, 
=  100,000, 

$378,000 

If  378,000  gain  6,000,  what  will    88,000,  A's,  gain? 
If  378,000     "        "         "       "    120,000,  B's,     " 
If  378,000    "        "         "      "    170,000,  C's,     " 


A  gains 
B     " 


$1,396.83,  ) 

$1,904.76,  \  =6,000,  PROOF, 

$2,698.41,  ) 


RULE    FOR   THE    FELLOWSHIPS.  181 

From  the  examples  given,  we  deduce  the  following 

SUMMARY   OP   DIRECTIONS   FOR   SIMPLE   FELLOWSHIP. 

Add  the  several  sums:  make  the  whole  sum  the 
supposition ;  each  separate  sum,  the  demand ;  and  the 
whole  gain  or  loss,  the  term  of  answer :  the  answer 
will,  in  each  case9  be  the  gain  or  loss  on  the  individual 
sums. 

PROOF:  Add  the  several  answers,  and  the  whole 
sum  will  equal  the  whole  gain  or  loss. 

To  make  General  Average, 

FIRST  :  Ascertain  the  value  of  the  cargo  ;  the  sum 
of  the  freight-Nil,  passage,  etc.,  minus  one-half,  or 
one-third,  as  customary :  the  sum  total  will  be  the 
general  value : 

SECOND  :  Ascertain  the  sum  of  loss,  goods  throivn 
overboard ;  cost  of  new  masts,  spars,  rigging,  etc.y 
minus  one-third;  commission  on  repairs;  port  duties, 
and  other  expenses :  add  these,  and  the  sum  total  will 
be  the  loss.  Then,  make  each  individual  loss  the  de- 
mand; the  sum  of  all  the  combined  interests,  the  same 
name ;  and  the  whole  sum  of  loss,  the  term  of  answer. 

PROOF  :  The  several  answers,  added,  will  equal  the 
whole  loss. 

COMPOUND    FELLOWSHIP. 

Multiply  each  sum  of  capital  by  the  time  invested; 
add  the  products ;  make,  each  separate  product  the 
demand;  the  sum  of  the  products,  the  same  name; 
and  the  whole  gain  or  loss,  the  term  of  answer. 

PROOF  :  Add  the  several  specific  sums,  and  they  will 
equal  the  whole  sum  of  gain  or  loss. 

When  a  part  of  an  investment  is  deducted,  or  an 
additional  sum  paid  in,  make  the  remainder,  or 


182  RAINEY'S  IMPROVED  ABACUS. 

the  increased  sum,  as  the  c-tse  may  be,  a  new  invest- 
ment for  the  time  that  it  runs ;  and  multiply  by  the 
time,  as  before. 

When  tzoo  or  more  investments,  made  by  one  indi- 
vidual, are  multiplied  thus,  the  separate  products  may 
be  added  into  one  individual  product,  before  being 
placed  on  the  line. 


BARTER,  DUTIES,  AND  COMMERCIAL  EX- 
CHANGE. 

Barter  *  is  the  exchange  of  one  article  of  specified 
value,  for  an  equivalent  value  in  something  else. 
Hence,  articles  in  Barter,  or  Commercial  Exchange, 
are  considered  in  the  ratio  of  their  separate  values  or 
quantities. 

How  many  bushels  of  wheat  at  50  cents  per  bushel, 
will  pay  for  200  bushels  of  corn,  at  30  cents  per 
bushel? 

The  demand  here,  is,  what  will  200x30  cents  buy, 
if  50  cents,  opposite,  buy  1  bushel?  Thus 

The  answer  is  120  bushels  of  wheat. 
The  30  and  200  are  here  multiplied, 
j2Q  ,  merely  to  show  that  the  demand  is  the 

price  of  the  whole  of  the  corn ;  other- 
wise the  proportion  would  not  be  recognized.  It 
should  be  stated  with  the  terms  to  be  multiplied,  one 
jvhove  the  other;  thus, 

*  Barter  is  from  the  Spanish,  baratar,  from  the  Latin  root,  verto,  to  turn 
or  exchange.  This  change  implies  equality  in  the  articles,  or  the  prices  of 
the  articles,  exchanged.  From  its  derivation,  it  hears  a  striking  resem- 
blance to  the  word  proportion. 


BARTER. 


183 


One  hundred  and  twenty,  as  before. 


200 
5030_ 

120 

How  many  bushels  of  corn,  at  30  cents  per  bushel, 
will  pay  for  120  bushels  of  wheat,  at  50  cents  per 
bushel? 


The  answer  is  200  bushels  of  corn. 


50 


|200 


If  200  bushels  of  corn,  at  30  cents  per  bushel,  pay 
for  120  bushels  of  wheat,  what  is  the  price  of  the 
wheat  ? 

r—Wp  £00 

£0—5 


The  answer  is  50  cents. 


50  cts. 

How  many  pounds  of  butter,  at  12^  cents  per  lb., 
will  pay  for  1&|  Ibs.  of  bacon,  at  10  cents  per  lb.? 


Here,  the  demand  is  18f  times  10 
cents ;  the  same  name  12-J-  cents ;  and 
the  term  of  answer,  1  pound  of  butter : 
hence,  15  Ibs.  of  butter,  the  answer. 


15  Ibs. 


How  many  pounds  of  sugar,  at  3-J-  cents  per  lb., 
will  pay  for  20  bbls.  of  rum,  35  galls,  to  the  barrel, 
worth  90  cents  per  gallon  ? 


The  answer  is  18,000  pounds  of  sugar. 


20 

tl> 
90 

2 


l>—  5 


18,000 
If  18,000  Ibs.  of  sugar  pay  for  20  bbls.  of  ram 


184 


RAINEY'S  IMPROVED  ABACUS. 


35  gallons  to  the  barrel,  and  worth  90  cents  per  gal- 
lon, what  is  the  sugar  worth  per  lb.? 


What  will  1  lb.  of  sugar  cost, 
if  18,000  Ibs.  cost  20x35x90 
cents  ? 


If  18,000  Ibs.  sugar,  at  3£  cts.  per  lb.,  pay  for  20 
bbls.  of  rum,  worth  90  cents  per  gallon,  howrnany  gal- 
lons are  there  per  barrel  ? 

>— 5 


It  is  seen  in  this  and  the  preceding 
examples,  that  all  of  the  constituents 
of  the  given  commodity  are  placed  on 
or      I!  the  right,  and  the  remaining  constitu- 

ents, of  the  commodity  about  which 
the  inquiry  is  made,  on  the  left.  If  there  are  four 
constituents  on  the  right  for  the  perfect  supposition, 
and  three  out  of  four  given,  for  an  equivalent  in  barter, 
the  three  given,  must  be  placed  on  the  left,  and  the 
other  factor  or  constituent  will  be  found.  It  is  ob- 
served that  the  price  is  placed  on  the  left,  to  find  its 
associated  quantity ;  or,  the  quantity  is  placed  on  the 
left,  to  find  its  associated  price. 

BARTER   BY   REDUCTION 

All  reduction,  ascending  and  descending,  is  a  spe- 
cies of  concatenated  or  conjoined  proportion.  Where 
articles  are  of  different  denominations,  or  are  paid  for 
in  prices  of  different  denominations,  it  is  very  con- 
venient to  make  these  reductions  on  the  line,  at  the 
same  time  that  the  general  question  is  wrought.  And, 
if  treated  proportionally,  the  statement  becomes  lucid, 
and  the  work  interesting. 

How  many  yards  of  cloth,  at  $1,25  per  yard,  will 
pay  for  6  tons  of  iron,  and  5  pence  per  lb.,  in  the  cur- 


BARTER    BY    REDUCTION. 


185 


Ton.     1 
Cwt.     1 
Lb.       1 
D.      12 
Sh.       6 
Cts.  125 

6       Tons. 
20     Cwt. 
100  Lbs. 
5       Pence. 
1       Shil. 
100  Cents. 
1       Yard. 

A  ns. 

666f  yds. 

rency*  of  New  England,  which  is  6  shillings  to  the 
dollar? 

What  will  6  tons  make,  if  1 
ton  make  20  cwt.;  what  will  this 
make,  if  1  cwt.  make  100  Ibs.; 
what  will  all  of  these  Ibs.  come 
to,  if  1  Ib.  cost  5  pence;  how 
many  shillings  will  these  pence 
make,  if  12  pence,  opposite, 
make  1  shilling;  how  many 
cents  will  all  of  these  shillings 
make,  if  6  shillings,  New  England,  opposite,  make  100 
cents;  how  many  yards  will  all  these  cents  buy,  if 
125  cents  buy  1  yard?  Hence,  the  answer  is  in 
yards,  666-|;  the  number  that  will  pay  for  the  6  tons. 

Let  us  now  prove  it,  by  asking,  how  many  tons  will 
pay  for  666-|  yards.  The  latter  number  becomes  the 
demand;  thus,  o 

Yd.       12000  Yds. 
Cts.  100125     Cts. 
Sh.        16         Sh. 
D.         512      D. 
Lbs.  1001         Lb. 
Cwt.    201         Cwt. 
Ton. 


The  ones  arc  wholly  unne- 
cessary in  the  solution;  and 
are  given  merely  to  indicate 
the  ratio 


Ans.\6  tons. 


beautiful  decimal  system.    Sterling  money  derived  its  name  from  Ester 
ling,vfho  first  made  the  coin.     The  dollar  mark  is  a  combination  of  U.  S, 

nnrl   \vn«  nrirrinallv  \vrittpn    TT     S    Oft  OO.  Pt<v 


and  was  originally  written  U.  S.  20.00,  etc 


186 


RA1NEY  S    IMPROVED    ABACUS. 


We  may  prove  it,  again,  by  finding  the  cost  of  1  Ib. 
of  iron,  which  was  5  d.;  hence,  the  5  d.  is  the  demand : 
D.         125       D. 
Sh.         6 1       Sh. 
Cts.    125100  Cts. 
Yds.  2000 1      Yd. 

3 
Ton.        1 6       Tons. 

20     Cwt. 

100  Lbs. 


Denominations  of  the 
same  kind  are  placed  oppo- 
site each  other. 


Ans.  1  Ib.  iron. 
This  may  be  proven,  by  demanding  cost  of  1  Ib.  iron  : 


This  might  be  proven  by 
several  other  processes;  but 
further  proof  is  unnecessary. 


Ans.  5  pence. 

How  many  galls,  cordial  at  $180  per  gal ,  will  pay 
for  10  T.  wine,  at  6  d.  per  pt.,  S.  Carolina  currency  ? 
10 


Lbs.  100 

1    Lb. 

Cwt.  20 

1    Cwt. 

Tons.   6 

1    Ton. 

3 

Yd.    1 

2000  Yds. 

Cts.  100 

125  Cts. 

Sh.    1 

6    Sh. 

12   D. 

4^030.00 


Ans.  1200  gallons. 


Tun.        1 

10        Tuns. 

Pipe.       1 

2         P. 

Hhd.       1 

2          Hhds. 

Gal.         1 

63        Gals. 

Qt.         1 

4          Qts. 

Pt.         1 

2         Pts. 

D.         12 

6          D. 

Sh.       20 

1         Sh. 

£.           7 

1         £. 

Cts.    1.80 

30.00  Cts. 

1          Gal. 

Ans. 

1200  gallons. 

BARTER    AND    COMMERCIAL  EXCHANGE.         187 


The  £7,  above  equal  $30,  or  3,000  cents,  South 
Carolina  currency. 

In  this  question,  the  demand  is  10  tuns;  the  same 
name,  1  tun :  and  the  term  of  answer,  the  next  lower 
denomination,  2  pipes.  Thus,  the  answer  of  every 
preceding  statement,  is  made  the  demand  of  a  subse- 
quent question. 

This  question  may  be  proven  in  several  ways,  as  the 
one  preceding. 

How  many  barrels  of  wheat,  at  50  cts.  per  bushel, 
will  pay  for  12  tons  of  iron,  at  4  pence  per  lb.,  New 
York  currency  ? 


1 
1 
1 

1-6 

W 
&fi 
100 

A 

/Ljl 

W 
* 

yk 
1 
400—4 
1 
1 

Ans. 

400 

Ton.     1 

12     Tons. 

Cwt.     1 

20     Cwt. 

Lb.       1 

100  Lbs. 

D.       12 

4      D. 

Sh.        8 

1       Sh. 

Cts.    50 

100  Cts. 

Bu.       5 

1       Bush. 

1       Barrel. 

Ans. 

400  barrels. 

In  New  York,  8  shillings  make  1  dollar ;  hence,  the 
8  on  the  left  of  the  line. 

From  the  foregoing,  we  deduce  the  following 

DIRECTIONS    FOR   BARTER,    COMMERCIAL    EXCHANGE,    ETC. 

Place  all  of  the  terms  of  the  commodity  whose 
quantity  and  value  are  given,  on  the  right ;  and  all 
of  the  terms  of  the  commodity  whose  quantity  or  value 
is  required,  on  the  left :  the  answer  will  be  the  quan- 
tity or  value,  as  the  case  may  be. 

FOR  DENOMINATE  NUMBERS:  Place  the  demand 
first  on  the  right :  place  a  unit  of  the  same  name,  or 
the  quantity  specified  of  the  same  name,  opposite  : 
place  the  number  which  this  unit  makes  in  reduction, 


188  RAINEY'S  IMPROVED  ABACUS. 

or  such  other  quantity  of  something  else  as  the  given 
quantity  may  equal,  on  the  right :  place,  again,  oppo- 
site this  last  term,  on  the  left,  the  unit  or  other  quan- 
tity of  the  same  name,  and  proceed  as  before ;  mak- 
ing the  answer  of  each  preceding,  the  demand  of  a 
succeeding  question,  ad  infinitum. 


DUTIES,  TARE  AND  TRET,  ETC. 

We  shall  bring  together  and  consider  in  one  article, 
both  Duties  and  Tare  and  Tret,  because  of  their  al- 
most entire  identity,  and  the  similarity  of  operations 
necessary  to  reduce  them ;  both  depending  alike  on 
per  centage,  as  well  as  losses  of  tare,  tret,  leakage,  etc. 

Commercial  duties  are  sums  of  money  paid  to  a 
government,  for  the  privilege  of  importing  foreign 
goods.  The  general  law  imposing  these  duties,  and 
determining  their  rate  and  extent,  is  called  a  tariff. 
Tariffs  are  different  in  different  countries,  varying 
in  their  extent  according  to  the  necessities  which 
cause  their  creation.  In  some  countries,  a  charge 
is  made  on  goods,  produce,  etc.,  for  the  ostensible 
purpose  of  creating  revenue  to  discharge  the  expenses 
of  government;  in  others,  for  the  purpose  of  pro- 
tecting manufactures,  or  agriculture,  by  preventing 
the  importation  of  foreign  manufactures  or  agricultu- 
ral products.  The  manufacturing  facilities  of  Great 
Britain  being  such  that  she  can,  not  only  compete 
with,  but  defy  the  whole  world,  she  lays  a  tariff  more 
particularly  on  the  products  of  the  soil,  both  for  the 
promotion  of  industry,  with  the  full  development  of 
her  natural  resources,  and  the  establishment,  among 

Tret  is  from  the  Latin,  tritiis,  from  tero,  to  wear  ;  Tare  is  French,  and  is 
from  the  same  root  as  the  Italian  tarare,  to  abate. 


DUTIES,    AND    TARE    AND    TRET.  180 

her  laboring  population,  of  a  long- cherished  system  of 
almost  arbitrary  tenantry. 

In  our  own  country,  to  the  contrary,  a  tariff 
is  imposed  for  the  avowed  purpose  of  making  the 
premium  on  foreign  privileges,  pay  the  expenses  of 
our  government.  Such  a  tariff,  varies  in  its  exac- 
tions, according  to  the  financial  exigencies  of  the 
government. 

We  have  had  a  tariff  in  our  country  for  quite  a  dif- 
ferent purpose ;  that  of  fostering  her  domestic  manu- 
factures. This  was  necessarily  high  enough  to  pre- 
vent a  large  foreign  importation,  by  rendering  the 
profits  accruing,  too  small  for  consideration. 

Our  federal  constitution  expressly  forbids  any  tariff 
between  the  several  states  of  this  union. 

1.  In  every  port  in  the  United  States,  where  vessels  are  permitted  to 
enter,  a  customhouse  is  established  by  the  government,  at  which  the  du- 
ties assessed  on  goods,  are  paid.     The  officer  who  examines  the  cargo, 
determines  the  duties,  etc.,  is  called  a  custom-house  officer.    All  importa- 
tions, whether  by  our  own,  or  foreign  vessels,  are   alike  subject  to  these 
duties. 

2.  Duties  are  divided  into  specific  and  ad  valorem. 

3.  A  specific  duty  is  a  certain  sum  charged  on  some  specified  quantity,  as 
a  cwt.,  gallon,  yard,  ton,  foot,  etc.     This  kind  of  duty  is  levied  without  re- 
ference to  price. 

4.  An  ad  valorem  duty  is  a  certain  per  cent,  on  the  price  of  the  article. 
This  phrase  is  from  the  Latin,  ad,  according  to,  and  valorem,  value  ;  ac- 
cording to  value. 

5.  When  specified  duties  are  received,  certain  allowances  are  made  for 
tare,  tret  or  draft,  leakage,  etc.    These  allowances  are  sometimes  a  speci- 
fied per  cent,  on  the  original  quantity  ;  and  sometimes  a  specified  deduc- 
tion on  the  cwt.,  cask,  box,  etc.;  and  are  deducted  before  the  duty  is  re- 
ceived ;  otherwise  duty  would  be  reckoned,  alike  on  the  quantity  wasted, 
and  that  saved. 

6.  Tare  is  an  allowance  made   for  the  box,  bag,  crate,  etc.,  which  con- 
tains the  article  ;  and  is  deducted,  either  specifically  or  at  a  certain  per 
cent.     It  has  been  customary  to  allow  12  Ibs.  on  the  112,  or  old  cwt. 

7.  Tret  is  a  certain  allowance  on  the  weight  after  tare  is  deducted,  for 
waste,  grease,  dust,  and  other  extraneous  substances,  and  is  usually  4  Ibs. 
on  the  104. 

8.  Leakage  is  an  allowance,  usually  of  2  per  cent.,  on  liquids,  such  as 
liquors,  oils,  chemicals,  etc..  for  waste. 

9.  Gross  weight  is  the  weight  before  any  deductions  are  mad«. 

13 


190 


RA1NEY  S   IMPROVED    ABACUS, 


10.  Suttle  is  the  weight  after  apart  of  the  deductions  are  made  ;  as  the 
Weight,  after  deducting  tare,  from  which  tret  is  to  be  deducted. 

11.  Net  weight  is  what  remains,  after  all  deductions  are  made  j  and  i» 
the  basis  on  which  duties  are  reckoned. 

12.  The  rates  of  less  are  different  on  different  articles,  according  to  th« 
law  regulating  such  deductions.    Losses  of  this  kind  are  sometimes  allow- 
ed in  individual  transactions,  in  groceries,  etc. 


SPECIFIC   DUTIES, 

What  is  the  specific  duty  on  1900  gallons  of  me- 
lasses,  at  9  cents  per  gallon,  allowing  2  per  cent,  for 
leakage  ? 

The  first  thing  to  be  done,  is  to  find  the  number  of 
gallons,  after  deducting  the  leakage.  What  will  1900 
be  reduced  to,  if  100  be  reduced  to  98 ;  thus, 

Here,  1900  gross  is  the  demand,  and 
100  gross,  the  same  name ;  while  98,  net, 
is  the  term  of  answer.  We  multiply 
the  net  number  of  gallons  by  9  cents, 
the  specific  duty,  and  find  the  duty, 
$167,58,  cutting  off  two  figures,  for 
cents.  We  should  have  placed  1  oppo- 
site, and  9  under  the  98,  saying,  what  will  all  of  the 
net  come  to,  if  1  gallon,  net,  be  9  cents? 

What  is  the  duty  on  20  hhds.  whisky,  at  10  cents 
per  gallon,  allowing  2  per  cent,  for  leakage  ? 


The  hhds.  are  reduced  to  gallons,  as 
in  other  cases 


At  6  cents  per  lb.,  what  is  the  specific  duty  on  170 
kegs  of  tobacco,  weighing  each  125  Ibs,  allowing  6 
Ibs.  per  hundred  for  tare? 


w 

1900 

98 

1862 
9 

$ 

167.58 

1 

100 

20 
63 

98 

$ 

123.48 

AD    VALOREM    DUTIES. 


191 


When  tare  is  so  much  per  box,  1  170 

bag,  etc.,  we  multiply  by  the  num-     £ — ,100  fL&p — 25 
ber  of  boxes,  bags,  or  whatever  1  94 

they  be :  find  the  whole  tare,  and  $ — 3 

after   subtracting   this   from    the  I  HQS  50 

whole  gross,  multiply  the  net  re- 
mainder by  the  specific  rate ;  thus, 

What  is  the  duty,  at  44  cents  per  lb.,  on  80  boxes 
tobacco,  weighing  each  100  Ibs.,  allowing  tare  at  20 
cents  per  box  ? 

Here,  the  tare  would  be,  at  20  Ibs.  per  box,  on  80 
boxes,  20x80=1600 Ibs.  Now,  8000-— 1600=6400 
Ibs.,  net.  This  net  weight  is  multiplied  by  the  spe- 
cific rate ;  thus, 


We  find  that  the  duty  is  $288,00. 
Thus,  it  is  seen,  that  we  can  use  the 
line  to  advantage,  even  in  specific 
duties. 


0^00—32 
9 


288,00 


DIRECTIONS    FOR    SPECIFIC   DUTIES. 

Find  the  net  weight,  and  multiply  this  by  the  duty. 
To  find  the  net  weight,  make  the  gross  sum  the  de- 
mand :  100  gross  the  same  name :  and  100,  di- 
minished by  the  tare,  tret,  or  leakage,  the  term  of  an- 
swer :  the  answer  will  be  the  net  weight.  Place  un- 
der the  term  representing  net  weight,  the  duty ;  cut 
off  two  figures  in  the  answer,  when  the  duty  is  in 
cents,  and  the  answer  will  be  in  dollars  and  cents. 


AD    VALOREM   DUTIES. 

It  may  be  again  observed,  that  ad  valorum  duties 
are  reckoned  on  the  cost  price  of  the  article;  hence, 
no  deductions  of  tare,  tret,  etc.,  are  made,  as  in  spe- 
cific duties.  The  duty  is  reckoned  on  the  invoice  of 
goods,  in  the  same  manner  that  premium  is  reckoned 
on  insurance. 


192 


RATNEY'S  IMPROVED  ABACUS. 


1.  An  invoice  is  a  bill  giving  in  all  the  goods,  and  setting  forth  the  price 
of  each  article. 

2.  To  prevent  deception,  this    bill  must  be  verified,  .or  sworn  to,  by  the 
owners,  or  one  of  the  owners  of  the  goods,  to  the  effect,  that  the  invoice 
presents  a  true  statement  of  cost  prices  ;  and  that  no  discount,  drawback, 
or  bounty  has  been  named,  that  has  not  been  actually  allowed. 

3.  This  oath  shall  be  administered  by  a  consul,  commercial   agent,  or 
other  duly  authorized  officer,  in  the  country  where  the    goods  are    pur- 
chased ;  which  fact  shall  be  duly  certified  by  such  consul  or  agent. 

4.  The  officer  thus  acting,  who  commits  any  fraud,  in  making  the  in- 
voice too  small,  etc.,  is  liable  to  heavy  penalties.  Our  government  keeps  a 
consul,  or  other   commercial  agent,  in  every  important  part  of  the  world, 
for  this  purpose,  except  where  they  are  forbidden  admittance  by  the  laws 
of  the  country,  as  has,  until  recently,  been  the  case  in  China. — See  Laws  of 
the  United  States. 

What  is  the  ad  valorem  duty,  at  20  per  cent.,  on  a 
lot  of  boots  and  shoes,  worth  $40,000? 

The  question  is,  what  duty  will  $40,000  give,  if 
$100  give  $20  duty?  thus, 

/100  40,000 
20 


$  8,000 


We  find  that  the  duty  is  $8,000. 


A  lot  of  Italian  silks  cost  $9,100 ;  wnat  is  the  ad 
valorem  duty  on  them,  at  43  per  cent.?  thus, 

400  9,100 
43 


$3,913 


The  answer  is  $3,913. 


What  is  the  duty  on  an  invoice  of  socks,  which  cost 
B,963,  at  25  per  cent.?  thus, 


4-400 


3963 


990f 


We  might  suspend  the  100  on 
the  left,  and  cut  off  two  figures  for 
decimals  of  a  dollar,  on  the  right. 


When  the  rate  is  the  aliquot  part  of  a  dollar,  we  may  very  much  abridge 
the  operation  by  suspending  the  100  on  the  left,  and  multiplying  by  such 
fractional  part  of  a  dollar,  as  the  rate  makes,  as  above  :  25  per  cent,  is  the 
%  of  100  ;  hence,  we  may  multiply  the  3963  by  £,  which  is  equivalent  to 
simply  dividing  it  by  4. 

At  22  per  cent.,  what  is  the  duty  on  $71,000  worth 
of  crockery-  ware  ? 


COMMERCIAL    TARE    AND    TRET. 


193 


This  answer  is  obtained  in  dollars. 


#071; 

22 


$  15,620 

What  is  the  duty  on  20  tons  of  wool,  at  30  cents 
per  lb.,  the  duty  being  60  per  cent.? 

The  first  point  is  to  know  how  much         1  20 
the  wool  comes  to  in  cents ;  hence,  the         1  20 
reduction    descending,  found    in    the 
statement.      After   being  reduced   to 
pounds,  it  is  multiplied  by  the  price 
per  lb.;  100  is  placed  opposite ;  and  the 
rate,  last  on  the  right. 


30 
60 


$  7200.00 


We  will  now  consider  tare  and  tret  in  a  different 
form.  In  ordinary  cases  coming  under  this  head,  12 
Ibs.,  on  the  112,  are  deducted  for  tare.  This  112  Ibs. 
was  the  old  standard  for  1  cwt.;  but  is  not  now  used. 
It  has  been  customary  to  deduct  4  Ibs.  per  104  for 
tret,  after  the  deduction  of  the  tare.  This  custom 
has  pretty  much  gone  out  of  use.  The  12  Ibs.  de- 
ducted for  tare,  should  be  deducted  from  112;  not 
from  100,  as  is  dishonestly  done  by  some  dealers. 
Deducting  12  from  the  100,  instead  of  112,  is  simi- 
lar in  principle  to  the  deduction  of  false  discount. 

In  this,  as  in  many  of  the  departments  of  arithme- 
tic, too  much  attention  is  given  to  questions  merely 
theoretical,  in  which  denominate  numbers  are  used, 
such  as  have  no  existence  in  business.  We  never  hear 
anything  of  Ibs.,  qrs.,  cwt.,  etc.,  in  business.  Hence, 
the  impropriety  of  introducing  such  questions  into 
a  practical  treatise. 

It  is  to  be  regretted  that  our  whole  system  of  de- 
nominate numbers,  with  the  exception  of  Federal 
Money  and  Avoirdupois  Weights,  is  so  varied  and 
irregular ;  while  the  decimal  substitute  could  be  made 
with  great  ease  and  palpable  advantage. 


194 


RAINEY7S    IMPROVED    ABACUS. 


We  subjoin  a  few  questions  involving  commercial 
tare  and  tret  : 

What  will  1  ton,  gross  weight,  of  hemp  come  to, 
at  $3,62-|  cents  per  gross  cwt.? 

We  here  use  2240  Ibs.  as  the  gross 
ton.  The  price  per  cwt.,  or  per  112  Ibs., 
gross?  being  in  cents,  the  answer  is  cents. 


725 


50 


Or,  what  will  20  cwt.  cost,  if  I  cwt.  cost  $2,62|? 
thus, 

120 


725 


The  answer  is  $72,50,  as  before. 


$  72,50 

What  will  8  tons  tobacco  come  to,  at  10  cents  per 
It).,  allowing  10  Ibs.  per  100  for  tare? 


8 

20 
90 
10 


The  answer  is  $1440,00. 


$  1440,00 

What  would  be  the  duty  on  the  same  at  37-J-  per 
cent.? 


8 


90 

|10 

75 


$  540,00 


Above,  1  ton  equals  20  cwt.,  and 
1  cwt.  equals  90  Ibs.,  net ;  while  1  Ib. 
net  equals  10  cents,  and  100  cts.,  give 
y  cents  duty. 


I  have  1500  Ibs.  of  cotton,  worth  5  d.  sterling  per 
Ib.,  net;  and  wish  to  know  the  duty,  if  I  allow  15 
Ibs.  per  100  for  tare,  and  4  Ibs.  per  100  for  tret,  the 
duty  being  40  per  cent.? 

We  reduce  the  5  d.  to  Federal  money,  by  using  £l 


COMBINATIONS    IN    TARE    AND    TRET. 


195 


on  the  left,  which  equals  $4,84  on  the  right;  or  by 
saying  that  20  shillings  equal  $4,84, 

WIMW-* 


This  combination  is  quite 
simple,  and  enables  us  to 
arrive  at  the  result  with 
very  few  figures. 


5— ££— -/ 


l^-9f-£ 
£—100484 


,$  49,36f 


Bought  200  hogsheads  of  sugar,  on  a  credit  of  12 
months,  weighing  each  1500  Ibs.,  gross,  for  which  I 
pay  2-j-  cents  per  lb.,  net,  the  deduction  for  tare  being 
5  per  cent.;  how  must  I  sell  it,  go  as  to  realize  80  per 
cent  profit,  being  allowed  5  per  cent,  discount? 

Here,  instead  of  saying 
in  the  discount,  that  105 
is  reduced  to  100,  and  using 
another  100  on  the  left,  from 
which  to  advance  to  80  per 
«ent.,  we  say  that  105  dis- 
count is  advanced  to  180 
profit. 


1 

jt00  4£00— 5 
1  #—13 


180 


11700,00 


What  would  the  sugar  be  sold  for,  in  Havre,  after 
deducting  20  per  cent,  for  duty,  from  the  prime  net 
value  ? 


400 

i 

£00 

4  £00—5 

t 

80 
180 

Hhd.       1 
G.       100 

Net.        1 
2 
Dis.    105 

M.  D.I  00 

200     Hhds. 
1500  Lbs.,  grosa. 

91       Net. 
5         Ots.  per  lb. 

80       Minus  duty. 
180     Profit. 

Am.  ft 

9360,00 

Above,  we  find  first  the  number  of  Ibs,,  then 


196  RAINEY'S  IMPROVED  ABACUS. 

duct  the  tare ;  find  the  cost  of  the  whole  at  the  price 
per  lb.;  deduct  the  discount,  and  instead  of  reducing 
the  105  to  100,  merely  reduce  it  to  80,  which  is  allow- 
ing for  the  20  per  cent.  duty.  It  is  not  necessary  to 
use  the  two  one  hundreds  which  come  between  the  105 
and  80,  in  which  105  is  reduced  to  100,  and  100 
again  to  80.  The  duty  being  now  deducted,  and  giv- 
ing the  net  cost,  we  advance  80  per  cent.,,  and  find  the 
selling  price,  which  is  $9,360.  This  answer  might  be 
obtained  in  francs,  by  placing  93  cents  opposite  the 
f  cents,  and  5  francs  on  the  right  under  the  2^-.  Al- 
though the  discount  would  then  be  made  on  the 
francs,  the  question  would  be  the  same ;  as  the  value 
would  not  be  changed. 

DIRECTIONS      FOR     AIX     VALOREM      DUTIES,      TARE      AND 
TRET,  ETC. 

To  find  the  ad  valorem  duty,  make  the  whole  sum 
the  demand ;  100,  the  same  name  ;  and  the  rate  duty, 
the  term  of  answer.  The  answer  will  be  the  duty  in 
dollars,  or  dollars  and  cents,  according  to  the  denomi- 
nation of  the  sum. 

Or,  Multiply  the  whale  sum  by  the  rate  duty,  and 
strike  off  two  or  more  figures  for  centsr  as  the  case 
may  be. 

FOR  TARE  ANI>  TRET  :  Place  the  sum  on  the  right ; 
the  standard  on  the  left ;  and  the  standard,  reduced  by 
the  tare,  on  the  right.  The  answer  will  be  the  net 
weight. 

FOR  DUTY:  Place  100  on  the  left,  and  the  rate 
duty  on  the  right :  the  answer  will  be  the  duty  in  the. 
denomination  of  the  sum. 

To  MAKE  DISCOUNTS,  PROFITS,  LOSSES,  ETC.:  Place 
10.0,  increased  by  the  rate  discount,  on  the  left ;  and 


COMMERCIAL    EXCHANGE. 


197 


100,  on  the  right:  also,  100  on  the  left,  and  100,  in- 
creased by  the  gain  per  cent.,  or  reduced  by  the  loss 
per  cent.,  on  the  right.  If  there  be  discount,  and  profit 
or  loss,  both  in  the  same  question,  suspend  the  two  one 
hundreds. 


COMMERCIAL   EXCHANGE. 

Operations  in  this  department  of  numbers  are  iden- 
tical with  those  of  Conjoined  Proportion;  hence,  the 
statements  are  simple  and  easy,  the  demand  being 
placed  on  the  right,  the  same  name  on  the  left,  and  its 
equivalent  on  the  right,  as  the  term  of  answer. 
Again,  this  term  of  answer  becomes  a  new  demand, 
while  a  term  of  the  same  name  is  placed  opposite,  and 
its  equivalent  in  value  again  on  the  right.  This  order 
is,  however,  frequently  interrupted,  by  the  introduc- 
tion of  discounts,  gains  and  losses,  reductions,  etc.; 
yet  such  may  be  easily  interwoven  and  combined  in 
the  statement  by  the  reflecting  student. 

Bought  2,200  Ibs.,  gross  weight,  of  wool,  and  was 
allowed  a  deduction  of  5  Ibs.  per  105  for  tret;  I  paid 
for  net  weight  3s.  6  d.  per  lb.,  New  York  currency, 
and  having  a  credit  of  12  months,  was  allowed  a  dis- 
count of  10  per  cent,  for  ready  money;  for  how  much 
did  I  afterward  sell  the  whole  to  gain  20  per  cent,  on 
my  investment-? 


100—  p 


G.     105 
Net.      1 
2 
Sh.       8 
Dis.  110 
Par.  100 

2200  Gross. 
100     Net. 
7         Shillings. 

1         Dollar. 
100     Par. 
120     Profit. 

«!P 

1,000 

Above,  what  will  2,200  gross  be  reduced  to,  if  105 


I(J8 


RAINEY'S  IMPROVED  ABACUS. 


gross  =  100  net;  and  1  Ib.  net  =  f  shillings;  and  8 
shillings  =  $1 ;  and  $110,  in  discount,  is  reduced  to 
$100  par;  and  $100  par,  opposite,  is  advanced  to 
$1*20 ,  20  per  cent,  profit  ?  In  the  solution,  the  two 
100s,  par  of  discount,  and  par  of  profit,  were  left  out ; 
and  110,  discount,  simply  advanced  to  120,  profit. 

Bought  1,500  Ibs.  of  butter,  at  7  d.  2  far.  per  Ib.,  New 
Jersey  currency,  and  was  allowed  30  Ibs.  per  100  for 
firkin,  and  4  Ibs.  per  100  for  impurities  or  tret,  t  had 
a  credit  of  1  year,  at  5  per  cent,  interest,  but  paying 
the  cash,  was  allowed  5  per  cent,  discount.  I  imme- 
diately sold  the  same  so  as  to  realize  60  per  cent, 
on  my  money  invested :  what  did  I  get  for  the  butter, 
in  Federal  money 


tt—W 


160 


G.    100 

1500 

Gross. 

Sut.  100 

70 

Suttle. 

Net.      1 

96 

Net. 

2 

15 

Pence. 

D.       12 

Sh.     20 

1 

Shilling. 

£'s.       3 

1 

£• 

Dis.  105 

8 

Dollars. 

Par.  100 

100 

Par. 

160 

Profit. 

$ 

128 

Ans. 

A  merchant  has  20,000  Ibs.  of  cotton,  which  he  can 
sell  a  t  4  d.  per  Ib.,  New  England  currency.  Fail- 
ing to  find  a  purchaser,  he  gives  to  A,  in  barter,  4-J- 
Ibs.  cotton  for  15  Ibs.  of  butter:  he  then  barters 
with  B,  giving  him  40  Ibs.  of  butter  for  3  yards  of 
gambroon :  again,  he  barters  all  of  his  gambroon  with 
C,  giving  him  2-J  yards  gambroon  for  2  yards  broad- 
cloth :  now,  he  barters  his  broadcloth  with  D,  giving 
him  1}  yards  for  12  yards  linen ;  and  to  E  he  gives 
30  yards  linen  for  8  cwt.  sugar :  he  now  barters  his* 
sugar  with  F,  giving  3  cwt.  of  sugar  for  50  gallons  of 


COMMERCIAL    EXCHANGE. 


190 


melasses :  after  this,  he  gives  G  4-^  galls,  melasses  for 
1^  galls,  of  rum:  he  gives  to  H  400  gallons  of  rum 
for  3  horses:  and,  finally,  to  J  he  gives  2  horses  for 
120  sheep:  he  sells  his  sheep  at  $1,80  cts.  per  head; 
how  much  is  he  gainer  or  loser  by  trading,  instead  of 
taking  the  original  offer  for  his  cotton? 

0^0,000 


Lbs.      9 

20,000  Lbs. 

2 

B.      40 

15         Butter. 

Ga.       5 

3           Gambroon. 

2 

C.         6 

2           Yds.  cloth. 

5 

L.       30 

12         Linen. 

Su.       3 
M.        9 

8           Cwt.  sugar. 
50         Galls,  mel. 

2 

2 

3           Galls,  rum. 

R.    400 

H.        2 

3           Horses. 

S.         1 

120       Sheep. 
180       Cents. 

$ 

48000,00 

1 

3—  /W 

20 

3 

#0000 

4 
1 

1 

10,00 

Ans.  $ 

1111,111 

050 


,140 


Ans.  $48000,00 


1111,111 

Ans.  $46888,88| 

In  the  last  calculation,  £3  equal  $10  ;  hence,  the  an- 
swer above;  which,  subtracted  from  the  amount  ob- 
tained in  exchange,  leaves  for  the  gain  of  the  merchant 
in  trading  $46,888,88f  cents.  The  answer  of  the  * 


200  RAINEY'S  IMPROVED  ABACUS. 

first  question  is  cents,  because  the  price  of  one  sheep, 
last  on  the  line,  is  cents. 

From  the  foregoing  we  deduce  the  following 

DIRECTIONS   FOR    COMMERCIAL   EXCHANGE. 

Make  the  gross  quantity  the  demand;  the  specific 
quantity  the  same  name;  and  the  specific  quantity  of 
the  article  which  it  equals,  the  term  of  answer :  repeat 
the  process,  and  continue  the  concatenation  of  state- 
merit,  until  tht  last  term  of  answer  is  placed  on  the 
right ;  and  the  answer  will  be  in  the  denomination  of 
such  last  term. 

When  tare,  tret,  or  other  per  cent.,  is  to  be  deducted, 
place  the  standard,  whatever  it  be,  on  the  left ;  place 
the  same  standard,  reduced  by  the  deduction,  etc.,  on 
the  right,  for  suttle,  net  weight,  etc. 

When  a  discount  is  to  be  deducted,  place  the 
amount,  100  and  rate,  added,  on  the  left;  and 
100,  on  the  right,  for  present  worth: 

When  a  given  per  cent,  is  to  be  gained  or  lost, 
place  100  on  the  left,  and  100,  increased  by  the  gain, 
or  reduced  by  the  loss  per  cent.,  on  the  right ;  and  the 
answer  will  be  the  advanced  or  reduced  price. 


DECIMAL  FRACTIONS. 

Before  commencing  the  article  on  Mensuration,  it  may  be  well  to  give 
some  few  general  remarks  on  decimal  fractions,  that  the  pupil  may  be  pre- 
pared to  use  them  intelligibly,  as  they  constantly  occur  in  this  department 
of  numbers. 

Decimal  fractions  being  of  little  service  to  the  ordinary  arithmetician, 
except  in  Multiplication,  Division,  Addition,  and  Subtraction,  we  shall 
give  only  such  an  outline  of  their  nature  and  relations  as  will  meet  the 
wants  of  the  practical  calculator.  Consequently,  the  reader  will  not  look 
for  an  elaborate  explanation  of  abbreviations  in  decimal  calculations;  of 
circulating  decimals  ;  or  even  of  the  four  divisions  mentioned. 

Units   are   divided   into    regular    and    irregular 


THEORY    OF    DECIMAL    FRACTIONS.  201 

fractions.  When  divided  into  3ds,  4ths,  9ths,  IGths, 
etc.,  they  are  called  irregular  or  common  fractions; 
having  such  denominator  as  indicated  by  the  number 
of  equal  parts  into  which  the  unit  is  divided. 

In  decimal  fractions  the  unit  is  divided  into  ten 
equal  parts ;  while,  again,  one  of  the  latter  is  divided 
into  10  parts,  making  tenths,  hundredths,  thousandths, 
etc.  Hence,  the  name  decimal,  from  the  Latin, 
decem,  ten. 

If  units  increase  in  a  tenfold  ratio,  from  right  to 
left,  certainly,  from  left  to  right,  they  decrease  again 
in  the  same  ratio.  Now,  continuing  this  decrease, 
from  the  units  place  to  the  right,  it  is  palpable,  that 
numbers  decrease  in  each  successive  order,  in  the  ratio 
of  .J-,  yi-,  ToVo-»  etc.,  without  limit.  Hence,  the 
first  figure  to  the  right  of  units,  is  ten  times  smaller 
than  units;  the  second  one  hundred  times  smaller; 
the  third,  one  thousand  times  smaller,  and  so  on. 

The  point  ( , )  is  placed  between  whole  numbers  and 
decimals  to  distinguish  them ;  and  is  called  the  sepa- 
ratrix  or  decimal  point..  The  denominators  of  the  de- 
cimals .3,  .4,  and  .07,  would  be  T3¥,  T4¥,  T°/^.  Hence,  if 
the  decimal  numerator  belong  to  an  order  of  decimals 
below  tenths,  a  sufficient  number  of  ciphers  must  be 
prefixed  to  such  numerator,  to  supply  the  place  of  the 
vacant  orders.  In  the  case  of  yf^,  above,  it  is  neces- 
sary, in  showing  that  the  7  occupies  the  hundred's 
place,  to  place  a  cipher  before  it,  to  fill  the  tenth's  place. 
If  the  7  were  y^o^,  ^ree  ciphers  would  be  placed  at 
the  left  for  this  purpose,  and  would  be  written,  thus, 
.0007,  with  the  cipher  prefixed  as  far  as  the  order  of 
tens.  From  this  we  see,  that 

Tke  denominator  of  any  decimal  fraction  is  a  unity 
with  as  many  ciphers  annexed,  as  there  may  be  figures 
in  the  numerator. 

This  is  reasonable  when  we  reflect  that  the  denomi- 


202      RAINEY'S  IMPROVED  ABACUS. 

nator  of  each  separate  figure  in  the  numerator,  is  ten. 
Consequently, 

There  must  be  one  figure  less  in  the  numerator, 
than  in  the  denominator. 

If  it  be  necessary  to  express  three,  seven  hundred 
thousandths,  we  know  that  as  there  are  six  figures  in 
the  denominator,  there  must  be  five  in  the  numerator ; 
and,  as  the  3  belongs  to  the  order  of  hundreds  of 
thousandths,  the  five  vacant  orders  must  be  filled  by 
figures  that  express  no  value ;  thus,  .00003.  Hence, 

Ciphers  prefixed  to  decimal  digits,  have  no  active 
value,  and  serve  only  to  show  the  order  occupied  by 
such  digits. 

The  orders  of  decimals  and  their  names  may  be 
seen  in  the  following 

DECIMAL   TABLE. 


J         «   -2          »•    o         «    °    „•» 

1     »•  .3  sM     11     1 1  . « 1 1 

5     *  *  I  is     •  I     ^  *     * 

'g    tl   "a    o    a   Is  5    c'cs    e   la   3    all 

^       Q>       &     M       *>       B  i£       ®       3     73       «     _3       *-       W£ 


5347.457923865497329 

Here,  at  the  left  of  the  separatrix,  we  have  7  units,  4  tens,  3  hundreds, 
etc.;  while,  at  the  right  of  it,  we  have  4  tenths,  5  hundredths,  7  thou- 
sandths, etc.;  or  .45  hundredths;  .457  thousandths,  etc.  Again,  we  pass 
by  or  suspend  the  4  and  5,  and  say  that  the  792,  being  of  the  order  of  hun- 
dred thousandths,  taken  together,  may  be  expressed,  thus,  .00792;  the  two 
ciphers  prefixed,  showing  the  order  to  which  the  792  belong.  If  these  two 
ciphers  were  left  out,  the  792  would  be  so  many  thousandths  only. 

Three,  decimally  expressed,  would  be  .3,  or  -fa; 
but,  with  a  cipher  prefixed,  it  would  be  .03,  or  T£¥. 
Hence, 

Each  move  of  a  decimal  one  place  from  the  deci- 
mal point  or  unit,  decreases  its  value  ten  tines. 


ADDITION    OF    DECIMALS.  203 

Ciphers  annexed  to  decimals  do  not  change  their 
value;  as  the  significant  decimals  occupy  still  the 
same  order  or  value  in  relation  to  the  unit's  place. 

In  decimal  fractions,  the  denominator  is  never  written 
or  expressed ;  and  is  only  understood.  A  great  ad- 
vantage in  the  use  of  decimals  is,  that  instead  of  mul- 
tiplying or  dividing  by  the  denominator,  as  many 
figures  may  be  cut  off,  as  there  are  tens  in  the 
denominator. 

We  give  a  few  examples  in  the 

ADDITION   OF    DECIMALS. 

Add  318.972;  4.38;  62.7895;  and  3412.013; 
thus, 


318.972 

4.38 
62.7895 
3412.013 


3798.1545 


All  of  the  units  in  the  whole  numbers 
are  written  in  a  column;  8,  4,  2,  and  2. 
At  the  right  of  this,  the  decimals  are 
written,  tens  under  tens,  hundreds  under 
hundreds,  etc.,  each  order  under  its  sep- 
arate column,  and  under  a  similar  order. 
We  add,  as  in  other  cases,  beginning  at  the  right,  and 
carrying  all  that  may  be  over  nine  to  the  next  figure 
at  the  left,  both  in  the  decimals  and  the  whole  num- 
bers. After  this,  the  separatrix  is  placed  in  the  sum, 
in  its  own  column,  under  the  similar  separatrices  in 
the  sums  above.  Therefore, 


TO  ADD  DECIMALS: 

Place  down  the  several  whole  numbers  and  deci- 
mals, units  under  units;  tenths  under  tenths;  hun- 
dredths  under  hundredthst  etc.:  add  as  in  whole  num- 
bers, and  place  the  separatrix  of  the  sum  under  the 
separatrices  above. 


204  RAINEY'S  IMPROVED  ABACUS. 

SUBTRACTION  OF  DECIMALS. 

Subtraction  in  decimals  is  performed  as  in  case  of 
whole  numbers.  Let  the  smaller  number  be  written 
under  the  larger;  units  under  units;  tenths  under 
tenths ;  hundredths  under  hundredths,  etc. 

From  972.3856  subtract  298.534;  thus, 


972.3856 
298.534 

673.8516  i  n /»       10  o       i  i 

1  9  from  12,  3,  and  so  on ;  borrowing  as  in 

the  subtraction  of  integers.     Hence, 


Nothing  being  under  the  6  at  the  right 
hand,  we  say,  0  from  6  leaves  six :  4  from 
5,  one ;  3  from  8,  five ;  5  from  13,  eight ; 


TO   SUBTRACT    DECIMALS: 

Place  the  smaller  of  the  two  numbers  under  the 
larger ;  units  under  units ;  tenths  under  tenths;  hun- 
dredths under  hundredths,  etc.:  subtract  as  in  whole 
numbers,  and  locate  the  separatrix,  as  in  addition  of 
decimals. 

If  there  be  a  larger  number  of  decimals  in  the  lower 
than  in  the  upper  number,  ciphers  may  be  annexed,  ad 
infinitum,  to  the  decimal  in  the  upper  number.  We 
have  seen  before,  that  ciphers  thus  added,  do  not 
change  the  value  of  the  number  of  decimals  to  which 
they  are  appended. 

MULTIPLICATION    OF    DECIMALS. 

Multiply  .46  by  .5 

We  proceed  as  in  ordinary  multiplication ;  thus, 

.46    '       If  a  whole  number  be  multiplied  by  a  deci- 
.5 


.230 


mal,  the  answer  will  be  a  whole  number  and  a 
decimal  combined ;  if  a  decimal  be  multiplied 
by  a  decimal,  the  product,  according  to  the 
laws  of  multiplication,  must  be  decimals  only;  for 
decimal  factors  cannot  produce  integers ;  nor  can  in- 
tegers produce  decimals ;  that  is,  the  product  must  be 


MULTIPLICATION    OF    DECIMALS.  205 

of  the  denomination  of  the  multiplicand  and  the  mul- 
tiplier. Hence,  as  both  of  these,  in  the  question 
above,  are  decimals,  the  product  must  be  decimals ; 
and  as  there  are  no  integers,  there  can  be  no  integers 
in  the  result.  Therefore, 

As  many  figures  must  be  cut  off  for  decimals^  as 
there  are  decimal  factors,  or  places  in  both  the  mul- 
tiplicand and  multiplier. 

As  ciphers,  appended  to  decimals,  have  no  value,  the 
cipher  in  the  result  above  may  be  dropped,  and  the  de- 
cimal called  .23  hundredths,  instead  230  thousandths, 
which  is  equivalent,  as  before. 

Multiply  275.437  yards  of  cloth  by  3.07  dollars  per 
yard;  thus, 


Here,  both  the  multiplicand  and  mul- 
tiplier have  both  integers  and  decimals : 
hence,  there  are  both  integers  and  deci- 
mals in  the  answer.  Cutting  off  five 
places  for  decimals,  the  answer  is 
$845.59159 ;  or  59  cents  and  159  thou- 
sandths of  a  cent.  Hence,* 


275.437 
3.07 

1928059 
826311 

845^69159 


*SHORT    METHOD    OF    MULTIPLYING   DECIMALS. 

In  cases  where  the  multiplier  is  10,  100,  1000,  10000,  etc., 
the  decimal  point  may  be  removed  as  many  figures  to  the  right, 
as  there  are  ciphers  in  the  multiplicand.  Thus:  Multiply 
198.7486  by  100.  Thus,  19874.86,  Ans. 

Decimals  below  the  4th  and  5th  orders,  are  so  small  as  to  be 
of  very  little  value  :  hence,  when  the  decimal  places  are  very 
numerous  in  the  multiplicand  and  multiplier,  or  either,  and  it 
is  not  desired  to  extend  the  calcu\ation  beyond  4,  5,  or  6  deci- 
mal places,  it  becomes  necessary  to  resort  to  a  method  of  find- 
ing the  product  without  multiplying  to  whole  number  of  de- 
cimals. This  is  done  as  follows:  We  assume  the  following, 
in  ordinary  multiplication: 

The  left  hand  figure  of  the  multiplier  may  be  multiplied  by 

14 


RAINEY'S  IMPROVED  ABACITST. 

TO   MULTIPLY   DECIMAL    FRACTIONS. 

Proceed  as  in  the  multiplication  of  whole  numbers  / 
cutting  off  as  many  figures  in  the  product  for  deci- 
mals^ as  there  may  be  decimal  factors  or  places,  both 
in  the  multiplicand  and  multiplier. 

When  there  is  a  larger  number  of  decimal  places 


first,  If   tens  are  still  placed  under  tens;  .hundreds  under 
hundreds,  etc.;  thus, 


1284 
2475 


2568 
5136 
8988 
6420 


3177900 


1284 
2475 

2568 
5136 
8988 
6420 

8177900 


Hence,  as  above,  commencing  with  the  left  hand  figure  ID 
the  multiplier,  and  causing  the  scale  to  descend  te  the  right, 
produces  the  same  result,  as  multiplying  first  by  the  unit's 
place  and  descending  to  the  left. 

Now,  it  is  evident,  that  we  may  multiply  in  cases  of  deci- 
mals in  the  same  way,  and  by  carrying  the  multiplication  to 
a  certain  number  of  orders  to  the  right,  to  get  the  product  of 
the  orders  so  used,  throw  away  all  useless  and  minute  mul- 
tipliers. 

Let  us  multiply  2.8724  by  .37854;  thus, 


All  of  the  figures  to  the  right  of 
the  vertical  line,  are  useless,  as  there 
are  five  places  of  decimals  on  the 
left  of  it,  being  as  many  decimals  as 
desired  in  the  product. 

In  placing  down  the  entire  pro- 
ducts, the  figures  on  the  right  of  the 
line  serve  to  show  what  numbers  are 
carried  to  the  first  place  of  decimals 
retained  on  the  left. 


2.872'4 
.3785  4 

86172 
20106 
2297 
143 
11 

8 
92 
620 
4896 

1.0873  1 

8296  Ans. 

MULTIPLICATION    OF    DECIMAL    FRACTIONS.     207 

in  the  multiplicand  and  multiplier,  than  in  the  pro- 
duct, prefix  ciphers  to  the  product  until  the  defi- 
ciency is  supplied. 


We  may  now  show  how  to  get  the  figures  on  the  left  of 
the  line,  without  having  to  make  those  on  the  right. 


We  multiply  the  first  right-hand  figure  of 
the  multiplicand,  by  the  left-hand  figure 
of  the  multiplier,  placing  the  product  un- 
der the  figure  thus  multiplied.  We  next 
multiply  by  the  second  decimal  multiplier, 
7.  Multiplying  this  into  4  would  necessa- 
rily cause  the  product  to  be  placed  one 
move  to  the  right  of  the  former  product, 
2;  and,  as  this  is  unnecessary,  we  multi- 
ply the  7  into  the  second  figure  from  the 
multiplicand,  2,  carrying  to  the  product 


2.872'4 
.3785  4 

~86172 

20107 

2298 

143 

11 


1.08731  Ans. 


the  nearest  number  of  decimals  which  this  7  and  the 
suspended  4  would  make.  Seven  times  4  making  28, 
nearly  3  decimals,  we  say,  7  multiplied  by  2  equals  14,  and  3 
added,  makes  17.  Hence,  the  7  is  placed  under  the  2,  at  the 
right,  and  1  carried  to  the  product  of  7  into  7,  which  makes 
49,  making  it  50,  and  so  on.  We  multiply  again  by  the  next 
decimal  multiplier,  8,  suspending  both  the  4  and  2,  at  the 
right  of  the  multiplicand,  and  carry  its  product  into  the  7 
above,  the  nearest  number  of  decimals  that  the  2  and  4  make. 
Thus,  8  times  7  are  56:  now,  8  times  the  former  2  are  16, 
and  8  times  4  are  32  ;  carrying  3  from  32,  to  16,  makes  19; 
very  nearly  2  decimals:  hence,  2  added  to  56  make  58;  the  8 
being  placed  under  the  right-hand  column,  and  the  5  carried 
as  before.  In  like  manner  we  proceed  next  with  the  5  and  4. 
After  multiplying  8  by  5,  we  carry  3;  because  5  into  7,  the 
suspended  order,  makes  35,  which  we  call  only  3  decimals, 
although  it  is  three  and  a  half,  allowing  this  half  over 
3,  for  the  deficit  in  previous  numbers,  where  the  number  of 
decimals  carried,  was  rather  too  large.  Thus,  the  numbers 
become  pretty  well  balanced. 

Above,  we  must  always  cast  off  one  figure  less  in  the  mul- 
tiplicand than  the  number  of  decimals  which  we  wish  to  re- 
tain: for  the  first  decimal  figure  of  the  multiplier,  when  mul- 
tiplied by,  would  otherwise  give  one  factor  too  many;  and 
consequently,  one  place  of  decimals  too  many  in  the  product. 
The  first  multiplier  will,  however,  be  multiplied  into  the  re- 
jected figure,  and  the  nearest  number  of  decimals  in  the  pro- 


In  this  division,  the  decimal  on  the  left, 
equals  or   neutralizes   one    decimal   on  the 


.208  RAINEY'S  IMPROVED  ABACUS. 

DIVISION  OF  DECIMAL  FRACTIONS. 

How  many  gallons  of  melasses,  at  .4  of  a  dollar  per 
gallon,  can  be  bought  for  .96  of  a  dollar  ? 

Here,  .4  equals  T\  ;  and  .96  equals  T9¥6o  •  We  now 
divide  the  latter  common  fraction  by  the  former; 
thus, 

$ — 24  |  The  answer  is  in  units  of  gallons. 
Now,  instead  of  dividing  24  by  this 
10  on  the  left,  it  is  quite  as  easy  to 
cut  off  one  figure  at  the  right  of  the 
24;  thus,  2.4,  and  the  4  is  understood  as  T\,  or  a  de- 
cimal. Hence,  it  may  be  divided  decimally ;  thus, 

.4)  .96 

24 

right :  hence,  the  remaining  decimal  becomes 

a  unit. 

In  multiplying  decimals,  the  product  must  have  as 
many  decimals  as  there  are  decimals  in  both  the  mul- 
tiplicand and  multiplier.  The  dividend  is  always 
equal  to  the  product  of  the  divisor  and  quotient; 
hence  there  must  be  as  many  decimals  in  the  divisor 
and  quotient,  taken  together,  as  there  are  in  the  divi- 
dend. Therefore, 

To  locate  the  decimal  point,  ascertain  the  differ- 
ence between  the  number  oj  decimals  in  the  divisor 
and  the  dividend ;  the  remainder  will  be  the  number 
of  decimals  to  be  cut  off,  in  the  quotient.  If  there  are 
not  as  many  decimal  places  in  the  quotient  as  the  dif- 
ference^ prefix  ciphers  to  the  quotient,  until  the  num- 

duct,  will  be  carried  to  the  product  of  the  first  figure  of  the 
multiplicand.  The  rejected  figure  is  in  the  column  of  the 
last  decimal  of  the  answer.  Hence,  count  to  the  left  from 
this,  and  locate  the  decimal  point  accordingly. 

This  subject  cannot  be  treated  elaborately  here;  as  it  be- 
longs to  Elementary  Arithmetic. 


DIVISION    OF    DECIMALS. 


209 


ber   in    the,    quotient    equals  the  number  in  the  dif- 
ference. 
Divide  .00954  by  3.08. 


3.08).00954(.003 
924 

~30 


The  answer  is  3  thousandths. 
By  annexing  ciphers  to  the  30,  the 
division  may  be  continued,  and  the 
quotients  placed  at  the  right  of 


the  .003. 

However  far  the  division  may  be  carried  in  this  example,  it  will  not  ter- 
minate. It  is  called  a  circulating  decimal.  The  division  has  been  con- 
tinned  on  to  fourteen  places,  giving  the  following, 

.00309740292207792 

When  the  number  of  decimals  in  the  dividend  and  divisor,  is  the  same, 
the  quotient  is  a  whole  number.  When  there  are  not  as  many  decimals 
in  the  dividend  as  in  the  divisor,  annex  ciphers  to  the  former  until  they 
are  equal. 

It  is  generally  very  useless  to  carry  decimal  calculations  further  than 
four  or  five  figures 

TO   DIVIDE   DECIMAL    FRACTIONS.* 

Proceed  as  in  the  division  of  whole  numbers  ;  and 

^CONTRACTION    IN   THE   DIVISION    OF   TRACTIONS. 

Division  of  decimals  may  be  very  much  abridged  when 
there  is  a  very  large  number  of  decimal  places  in  the  divisor, 
as  in  the  following  example. 

Divide  4.3125  by  3.2364,  retaining  four  decimal  places  in 
the  answer. 

Common  Method.  Contraction. 

3.2364)4.3125(1.3324 
32364 

10761 
9709 

1052 
971 

81 

_65 

16 


3.2364)4.3125 
32364 

(1.3324 

10761 
9709 

0 
2 

1051 
970 

80 
92 

80 
64 

880 
728 

16 
12 

1520 
9456. 

3 

2064 

210  RAINEY'S  IMPROVED  ABACUS. 

cut  off,  in  the  quotient,  a  number  of  figures  equal  to 
the  excess  of  the  decimals  in  the  dividend,  compared 
with  those  in  the  divisor.  If  the  number  of  decimals 
in  the  quotient  be  too  small,  prefix  ciphers  until  the 
number  equals  the  excess,  above  named. 

Reduce  the  decimal  .225  to  an  equivalent  common 
fraction  of  the  lowest  term. 

Subscribing  the  denominator,  .225  becomes  fWo- 
This  reduced  to  its  lowest  term  is  ¥\.  Hence, 

TO    REDUCE    DECIMALS    TO    COMMON    FRACTIONS. 

Cancel  the  decimal  point,  and  place  the  denomina- 
tor below  the.  given  decimal;  reduce  the  fraction  to  its 
lowest  term,  and  the  answer  will  be  an  equivalent 
common  fraction,  in  its  lowest  term. 

The  first  figure  of  the  quotient  is  1.  Now,  instead  of  an- 
nexing a  cipher  to  each  remainder,  and  thus  multiplying  it 
successively  by  10,  we  reject  at  each  separate  division  the 
right-hand  figure  of  the  divisor,  which  is  equivalent  to  divi- 
ding it  successively  by  10.  In  multiplying  the  last  unrejected 
figure  in  the  divisor,  6,  by  the  second  quotient  figure,  3,  mak- 
ing 18,  we  carry  1,  which  is  the  nearest  decimal  that  the  pro- 
duct of  the  3  and  the  rejected  4,  will  make.  The  decimal 
accession,  from  the  rejected  figures  of  the  divisor  is  consid- 
ered hi  each  subsequent  multiplication  and  division,  until  the 
4  required  decimal  orders  are  found  for  the  quotient.  In 
multiplying  the  3  in  the  divisor,  by  the  third  3  of  the  quo- 
tient, the  product  is  increased  to  11  by  the  accession  from  the 
last  two  rejec4ed  figures,  3  times  6  making  18,  and  3  times  4 
making  12,  the  sum  of  which  is  20,  or  2  decimals. 

If  the  divisor  has  more  figures  than  the  number  required 
in  the  quotient,  including  integers  and  decimals,  take  as  many 
on  the  left  of  the  divisor  as  required  in  the  quotient,  and  di- 
vide by  them,  as  in  other  cases. 

If  the  number  in  the  divisor  be  smaller  than  that  required 
in  the  quotient,  divide  as  ordinarily,  until  the  deficiency  is 
filled;  after  which,  contract  as  before. 

When  the  divisor  is  10,  10-0,  1000,  etc.,  remove  the  separa- 
trix  to  the  left,  as  many  places  as  there  are  such  ciphers;  and 
the  division  will  be  performed  without  further  reckoning. 


COMMON   FRACTIONS    REDUCED   TO   DECIMALS.  211 

Keduce  f  to  an  equivalent  decimal  fraction. 

We  multiply  the  numerator  and  the  denominator, 
each,  by  100;  because  the  product  of  two  tens  into 
the  numerator,  is  the  smallest  number  that  can  be  di- 
vided by  the  denominator,  4.  This  makes  f  £ -J-,  which 
divided  by  the  original  denominator,  4,  gives  -£-£-0 
This  T7o%-  =  .75;  for  always  dividing  the  product  of 
the  numerator  and  denominator  into  tens,  hundreds, 
«tc.,  by  the  same  denominator  thus  multiplied,  it  is 
evident  that  the  denominator  must  always  be  composed 
of  tens.  .Since,  therefore,  these  tens,  hundreds,  etc., 
in  the  denominator  of  a  decimal,  are  useless,  we  avoid 
the  process  of  getting  them,  and  simply  annex  ciphers 
to  the  numerator  of  the  common  fraction,  and  divide 
by  the  denominator,  until  an  exact  result  is  obtained, 
or  as  many  decimal  places  as  requisite ;  Thus, 


The  division  here  terminates  in  two  places       ^— 
of  annexed  ciphers.     Again:  ,75 

Reduce  4  to  a  decimal, 

Here  the  divisor  terminates  in  one  place  of   |     '  — 
annexed  ciphers.  ,i 

Beduce  '-|*  to  a  decimal  ;  thus 

8)200000000000000000000 


666666666666666666664- 

TO   REDUCE   A   COMMON   TO   A   DECIMAL   FRACTION. 

Append  ciphers  to  the  numerator,  and  divide  by  the 
Denominator  •,  until  the    denominator    terminates,   or 

*Tbis  is  called  a  repeating  decimal  ;  showing  that  although  the  process 
anight  be  repeated  ad  infinitival,  yet  the  true  result  would  never  be  ob- 
tained. Hence,  although  we  get  nearer  to  the  true  answer  at  every  step, 
yet,  we  would  never  get  it  entirely,  although  the  division  were  continued 
forever.  In  such  cases  the  division  need  not  be  carried  further  than  from 
four  to  six  places,  as  in  the  seventh  place  one  of  the  sixes  would  be  only 
^.  ___  &.  ___  a  very  minute  and  scarcely  conceivable  common  fraction, 

1   0  0  ('  0  0  0  0'  J 

iT.h»?  /'/MA-  iu;irk  is  appended  to  aho.w  that  it  is  still  .imperfect. 


212      RAINEY'S  IMPROVED  ABACUS. 

until  a  sufficient  number  of  decimals  is  obtained.  Cut 
off,  for  decimals,  in  the  quotient,  a  number  of  places 
equal  to  the  number  of  ciphers  annexed. 

When  the  figures  in  the  quotient  are  not  equal  to 
the  number  of  ciphers  annexed,  prefix  ciphers  to  the 
quotient,  until  the  deficiency  is  supplied. 

The  method  of  pointing  off  above,  will  appear  reasonable,  when  we  re- 
flect, that  every  cipher  annexed  to  the  numerator,  multiplies  it  by  10; 
hence,  after  it  is  divided  by  the  denominator,  the  quotient  will  be  ten 
times  too  large,  and  should,  consequently  be  divided  again  by  10.  This  is 
done  most  easily,  by  cutting  off  one  figure  toward  the  left.  The  same  rea- 
soning is  true  as  regards  annexing  two,  three,  or  more  ciphers,  and  increas- 
ing in  the  multiplication,  by  100,  1000,  etc.,  necessitating  a  division  of  the 
result  by  the  same  numbers.  Hence,  the  propriety  off  striking  off  a  num- 
ber of  figures  in  the  result,  for  decimals,  equal  to  the  number  of  ciphers, 
appended. 

To  reduce  compound  numbers  to  decimals,  Reduce  the  denom- 
inate number  to  a  fraction  of  the  denomination  required,  and 
this  fraction,  to  a  decimal. 

Reduce  5  shillings  3  pence  to  the  decimal  of  a  pound. 
5s.  =60 d;   and  60+3=63 d:  now,  £1=240  d:  henceJLSL 
of  a  pound  must  now  be  reduced  to  a  decimal,  thus, 

24)63000(.2625 
48 


150 
144 

60 

48 


If  we  divide  by  the  240,  there  would  be 
a  cipher  in  the  unit's  place;  but  dropping 
the  cipher  in  the  divisor,  we  have  no  unit 
in  the  quotient,  and  place  the  decimal  point 
at  the  left  of  the  answer,  which  is  .2625 


decimals  of  a  pound. 
120 
120 

Reduce  15  minutes,  30  seconds  to  the  decimal  of  an  hour. 
!5X'60==>900-)-30=930    seconds:    now,    1    hour   contains 
3600  seconds:  hence,  reduce  -^^g-  to  a  decimal;  thus, 

36)9300QOO(.258S33 

The  result  is  .258333.  This  is  a  repeating  decimal;  hence> 
the  calculation  is  discontinued  at  six  places. 

All  other  redactions  in  denominate  numbers  may  be  mada 
as  these, 


DEFINITIONS  IN  GEOMETRY.  213 

MENSURATION, 

OR 

PRACTICAL   GEOMETRY.* 

MENSURATION  is  that  department  of  the 
science  of  numbers,  which  treats  of  the  meas- 
urement of  lines,  superfices,  solids,  &c.,  and 
is  derived  from  measura,  measure. 

The  general  principles  and  laws  regulating 
this  department  of  the  science,  are  derived 
from  Geometry. 

Geometry  is  the  science  of  magnitude,  in  all 
its  various  forms  and  relations ;  and  is  divided 
into  practical  and  theoretical.  The  latter 
treats  of  those  portions  which  are  so  complex 
as  to  require  symbols  and  the  higher  mathe- 
matical formulae  for  their  illustration :  the 
former  treats  of  such  portions  only,  as  depend 
on  the  simple  relations  of  numbers,  as  mani- 
fested through  proportion. 

Geometry  is  from  7*7,  the  earth,  and  /if?po*>, 
measure,  and  primarily  signified  the  measure- 
ment of  the  earth. 

Many  of  the  laws  of  Geometry  are  demon- 
strated by  formulae  that  the  ordinary  reader 

*  It  may  be  remarked,  while  treating  of  superficial  measurement, 
that  Abacus  is  a  Latin  word  which  means  flat,  fn  the  primitive 
ages,  ail  calculations  were  made  by  the  Oriental  nations  on  boards 
covered  with  dust,  on  which  lines  and  signs  could  be  easily 
traced.  From  the  word  abak,  signifying  dust,  the  Greeks 
deduced  their  word  a@*g.  This  Abacus  used  by  the  Romans,  and 
Abax  by  the  Greeks,  was  a  large  board  with  transverse  lines 
drawn  on  it,  on  which  calculations  were  made  by  sundry  move- 
ments of  pebbles,  or  calculi.  Hence,  the  derivation  of  our  En- 
glish word  calculate,  from  cakulo,  which  is  from  calculus,  a  pebble. 
The  latter  word  is  from  the  Syriac  kalkai,  gravel. 
ft 


214  RAINEY:S  IMPROVED  ABACUS. 

would  not  comprehend ;  they  can,  however,  be 
made  quite  as  intelligible  by  the  manifest 
relations  and  deductions  of  common  sense, 
without  the  exercise  of  which,  all  formulae 
become  mere  mechanical  arrangements ;  being 
neither  appreciated  nor  understood.  Too 
many  writers  endeavor  to  teach  Mensuration 
by  the  introduction  of  Geometrical  signs  and 
reasonings ;  thereby  endeavoring  to  teach  a 
primary,  by  the  rules  of  a  secondary  science. 
Hence,  the  reason  of  so  many  failures  in  this 
study ;  and  hence,  the  mechanical  patch-work 
by  which  many  practical  men  make  such 
calculations. 

Practical  Geometry  is  divided  into  superficial 
and  solid.  Superficial,  which  is  from  superficies, 
the  surface,  the  outside,  fyc.,  relates  to  the  meas- 
urement of  surface,  which  has  extent  merely, 
without  bulk  ;  and  has  two  sides  given  to  find  the 
contents :  Solid  Geometry  relates  to  the  measure- 
ment of  bodies  or  magnitudes,  which  have 
length,  breadth,  and  thickness.  This  species  of 
measurement  is  generally  called  cubic.  A  cubic 
foot  of  timber,  is  a  foot  Io7ig,  a  foot  wide,  and 
a  foot  thick, or  12  inches  in  everyway:  hence, 
when  these  3  twelves  are  multipled  continu- 
ously, they  make  1728,  the  number  of  cubic 
inches  in  a  cubic  foot.* 

Cubic  is  from  the  Latin  cubicus,  from  cubus, 
a  die.  Hence,  cubic  is  a  congregation  of  par- 
ticles, forming  a  solid  mass  of  six  equal  sides. 

According  to  the  laws  of  Multiplication, 
concrete  objects  cannot  be  multiplied  together ; 

*  This  solid  foot,  or  1728  cubic  inches,  weighs  1000  ounus 
rain  water. 


THEORY  OF  MENSURATION.  215 

nor  can  concrete  objects  of  different  denomi- 
nations be  multiplied.  Feet  multiplied  by 
feet,  through  ratio,  will  give  feet;  but  feet 
multiplied  by  inches,  will  give  neither  feet  nor 
inches.  Ten  feet  long  and  12  inches  wide 
will  give  neither  120  feet  nor  120  inches. 
Hence,  when  the  denominations  are  dissimilar ', 
such  reductions  must  be  instituted  as  will  make  the 
terms  alike.  Above,  if  we  divide  the  width  12, 
by  the  number  of  inches  in  a  foot,  we  find  that 
the  width  is  1  foot;  now,  the  length  and  width 
being  in  feet,  we  conclude  that  there  are  10 
superficial  feet. 

Ten  feet  in  length  and  10  in  width  give  100 
feet  superfice :  this  multiplied  by  10  feet  in 
height,  will  give  1000  feet  solidity. 

These  similar  dimensions •,  length,  height  h,  and 
width,  multiplied  together,  give  the  cubic  or  solid 
contents  of  the  figure,  in  the  denomination  of  the 
dimensions :  as  a  crib,  a  box,  a  wall,  a  boat,  a 
cistern,  &c. 

Let  the  learner  keep  these  truths  before  his 
mind,  and  but  few  difficulties  will  present 
themselves  in  ordinary  measurements.  When- 
ever mathematical  rules  are  introduced,  they 
must  be  received  by  the  student  on  authority, 
as  it  would  be  impossible,  In  a  treatise  on 
numbers,  to  develop  their  principles. 

It  may  be  remarked  here,  that  too  much 
time  is  generally  spent  on  algebraic  and 
mathematical  solutions,  while  the  learner  pro- 
poses to  study  arithmetic.  The  introduction 
of  such  questions  and  rules,  is  an  oversight  in 
too  many  authors :  for  the  student  thus  wastes 
his  time  in  pursuing  the  work  of  mathematics, 


216  RAINEY'S   IMPROVED   ABACUS. 

which  is  impossible  in  arithmetic;  while  it 
should  be  devoted  to  numbers  only ;  for  noth- 
ing: else  than  numbers  can  be  learned  in 
arithmetic. 


WOOD    AND    BARK. 

WOOD  and  BARK  are  generally  measured  by 
the  cord,  which  is  a  pile  4  feet  wide,  4  feet 
high,  and  8  feet  long.  The  word  cord  is 
derived  from  the  Welsh  cord,  signifying  a  twist, 
relating  to  a  rope :  hence,  the  cord,  or  rope, 
with  which  the  ancients  were  accustomed  to 
measure  a  pile  of  wood,  gave  128  solid  feet, 
which  these  dimensions,  4,  4  and  8  make, 
when  multiplied.  A  pile  of  wood  contains 
more  or  less  than  a  cord,  when  it  has  more  or 
less  than  128  solid  feet.  Hence,  wood  is 
measured  by  proportion.  We  may  multiply 
together  the  dimensions  of  the  pile,  and  com- 
pare the  whole  number  of  feet  with  128;  or 
we  may  compare  the  several  separate  dimen- 
sions with  4, 4,  and  8.  The  latter  is  preferable. 

How  many  cords  of  wood  in  a  pile  120  feet 
long,  20  feet  wide,  and  2  feet  high? 

g.^rfx ir     i       Here,  we  place  the  several 


2— ££ 


4  £0—5 


|37-J-  cords. 


dimensions  of  the  pile,  on  the 
right,  and  the  dimensions  of 
a  cord  opposite  these,  on  the 


left ;  and  say,  what  will  all 
these  feet  on  the  right  make,  if  8,  4,  and  4, 
on  the  left,  make  1  cord,  last  on  the  right, 
it  is  unnecessary  to  place  the  1  on  the  right, 
as  it  will  not  assist  in  the  calculation. 


WOOD  AND  BARK  MEASURE. 


217 


How  many  cords  in  a  pile  200  feet  long,  3? 
feet  wide,  and  16  feet  high? 


Here,  we  say  4  times  4  on 
the  left,  equal  16  on  the  right. 
The  answer  is  eighty-seven 
and  a  half  cords. 


cords. 


— 7 


32|175 


51 1  cds. 


How  many  cords  in  a  pile  of  bark  20  feet 
long,  3  feet  4  inches  high,  and  10i  feet  wide? 

In  this  instance,  4  inches 
are  £  of  a  foot,  making  the 
height  3J  or  y>  feet.  The 
numerator  of  this,  as  well  as  j 
of  the  *y ,  is  placed  on  the 
right,  and  the  denominator 
opposite.  The  answer  is 
5i|  cords. 

What  will  a  load  of  wood  8  feet  long,  2  feet 
6  inches  high,  and  3  feet  4  inches  wide,  come 
to,  at  1  dollar  and  80  cents  per  cord. 

Again,  the  inches  are  made 
the  fractional  part  of  a  foot, 
and  added  to  the  given  feet 
in  each  case ;  while  the  mixed 
number  is  placed  on  the  line 
in  the  form  of  an  improper 
fraction.  We  know  that  the 
8,  4,  and  4,  on  the  left,  make 


5 

/1 0—5 


41375 


93|cents. 

one  cord,  or  that  these  128  feet  are  worth  the 
price,  180  cents;  then  the  price  is  placed  last 
on  the  right  in  the  place  of  the  one  cord,  and 
the  answer  must  be  the  price  of  the  whole 
pile  of  wood  at  180  cents  per  cord.  This  is 
nothing  more  than  simple  proportion.  Twice 


218 


RAINEY'S  IMPROVED  ABACUS. 


3  on  the  left,  goes  into  18  on  the  right  three 
times.  The  answer  is  93 J  cents.  This  method 
is  quite  preferable  to  ascertaining  the  quantity, 
which  may  be  fractional,  and  multiplying  it  by 
the  price  as  a  separate  operation. 

How  many  cords  in  a  pile 
of  wood  10  feet  long,  3  feet 
wide,  and  7  feet  high?  and 
what  will  the  same  come  to, 
at  240  cents  per  cord? 

There  are  1|1  cords  wood. 

We  will  now  state  both  in 
one,  thus.  The  two  was  used 
in  4  on  the  left,  and  in  10  on 


64J105 


9  |   3,93f 


2|75 


\r4JA.         o          r*      11JL    *    ULL     ILL 

_itt=tl6    the  right. 

What  will  a  pile  of  wood 
40  feet  long,  3  feet  high,  and 
20  feet  wide,  come  to,  at  2 
dollars  per  cord? 

The  price  is  dollars  in  this 
case,  and  the  answer  is  in 
dollars ;  37  dollars  arid  50 
cents. 

From  the  foregoing,  we  conclude  that, 
To  ascertain  the  number  of  cords  in  a  pile  or 
load  of  wood  or  bark,  place  all  of  the  dimensions 
on  the  right  in  feet,  and  4,  4,  and  8,  or  128,  on 
the  left.  If  there  are  inches  in  any  of  the  dimen- 
sions, they  must  be  reduced  to  the  fraction  of  a 
foot,  and  added  to  the  feet,  and  treated  as  other 
improper  fractions . 

If  the  answer  is  desired  in  the  price  of  the 
whole  quantity  of  wood,  place  the  price  of  one  cord 
last  on  the  right,  in  dollars  or  cents,  and  the 


BOARD  MEASURE.  219 

answer  will  be  the  price  of  the  whole,  in  dollars  or 
cents. 

LUMBER    MEASURE. 

Under  this  head  may  be  classed  superficial 
board  measure,  and  the  measurement  of  solid 
timber.  We  have  only  two  general  dimensions 
in  board  measure;  length  and  width.  The 
thickness  is  generally  considered  a  unit ;  inch 
measure  being  the  standard.  Anything  under 
one  inch  is  not  noticed ;  but  all  above  an  inch 
in  thickness,  as  two  inches,  three  inches,  &c., 
is  called  two,  three,  &c.,  thicknesses  of  lumber. 
If  a  piece  of  lumber  20  feet  long,  16  inches 
wide,  and  8  inches  thick,  be  measured,  the 
thickness  is  called  eight  planks.* 

The  first  thing  to  be  done  with  such  a  ques- 
tion as  the  one  above,  is  to  reduce  the  width, 
which  is  in  inches,  to  feet,  that  width  and 
length  in  feet  may  be  multiplied  together  for 
the  superficial  contents.  This  would  after- 
wards be  multiplied  by  the  8  thicknesses,  giving 
8  times  as  many  feet  as  in  the  one  piece.  All 
of  this  may  be  done  in  the  same  operation ;  thus, 

It  will  be  observed  here,  that 
the  length,  width,  and  thickness 
are  all  placed  on  the  right  of 
the  line,  and  12  only,  on  the 
left,  to  reduce  the  width,  16 
inches,  to  feet.  Hence,  the  an-  '  |213£  ft. 

swer  is  213J  feet. 

*  In  America  the  words  board  and  plank,  are  variously 
used  to  denote  the  same  thing.  This  is  incorrect.  While 
a  board  is  a  thin  piece  of  timber,  a  plank  is  a  thick  and 
heavy  piece.  The  word  is  from  the  Dutch  plank,  or  the 
Danish  planke,  a  thick  board.  Hence  the  difference. 


RAINEY'S  IMPROVED  ABACUS. 


8|135 


161  ft. 


When  it  is  necessary  to  get  the  cubic  con- 
tents, we  place  another  12  on  the  left,  to  reduce 
the  thickness  in  inches  to  feet,  that  having  all 
three  dimensions  in  feet,  the  product  may  be 
feet. 

How  many  feet  of  lumber  in  a  board  18 
feet  long, 7i  inches  wide,  and  li  inches  thick? 
Here,  the  mixed  numbers  are 
reduced  to  improper  fractions, 
as  in  all  other  cases.  No  other 
statement  is  necessary  in  board 
measure,  than  such  as  will  ad- 
mit of  the  several  dimensions 
being  multiplied  together.  Now 
the  width  being  generally  in  inches,  and  the 
thickness  often  fractional,  it  is  quite  conve- 
nient to  throw  the  numbers  on  the  line,  and 
the  standard  which  reduces  the  inches  to  feet, 
with  the  denominators,  on  the  left. 

<15  How  many  feet  in  a  board  7i 

$ ^       feet  long,  8  inches  wide,  and  4i 

0 — 3       inches  thick  ?     When  the  length 
~,        ,      is  a  mixed  number,  as  in  this  in- 
^    '     stance,  it  must  be  reduced  to  an 
improper  fraction,  as  in  other  cases. 

What  will  a  pile  of  planks  containing  120 
pieces,  16  feet  long,  15  inches  wide,  and  6 
inches  thick,  come  to,  at  37i  cents  per  100  ft.  ? 
In  this  instance,  we  mul- 
tiply by  120,  the  number  of 
pieces,  and  ascertain  the 
number  of  feet  in  the  whole 
pile :  the  question  is  then 
proportional,  and  by  com- 
bination of  statement,  we 


|  54,00 


MENSURATION. 


221 


say,  what  will  all  of  these  feet  come  to,  on  the 
right,  if  100  feet  opposite,  cost  37i,  or  y  cents  ? 
The  answer  is  5400,  the  number  of  dollars  and 
cents,  which  pay  for  the  whole. 


12 


16 


100 
275 


16 

15 

6 

120 

13 


Suppose  in  the  case  above  the 
timber  will  lose  T\  of  an  inch  in 
sawing.  We  say,  what  will  the 
whole  quantity  of  lumber  be  re- 
duced to,  if  y§  be  reduced  to  i-f  ? 

$|43,87i 

What  will  4  pieces  of  timber  come  to,  at 
$2i  per  100  ft.  which  are  10,  20, 18  and  12  feet 
long  respectively,  and  16  inches  wide,  and  3 
inches  thick  ? 

10 
20 
18 
12 

60  entire  length. 

In  this  instance,  it  is  necessary  to  add  the 
several  lengths,  and  place  their  sum  on  the 
right.  Had  there  been  10,  or  any  other  num- 
ber of  pieces  in  each  pile,  10  or  such  number 
would  be  placed  on  the  right,  once,  and  only 
once :  for  the  question,  by  getting  the  sum  of 
the  lengths,  was  changed  into  this,  how  many 
feet  in  a  piece  60  feet  long,  16  inches  wide, 
and  3  inches  thick  ?  Hence,  ten  times  the 
number  in  each  case,  would  be  ten  times  the 
60  feet. 

What  will  10  piles  lumber,  with  40  pieces 
lo 


222 


KAINEY'S   IMPROVED   ABACUS. 


in  a  pile,  come  to,  at  $H  per  100  feet,  the 
plank  being  18  inches  wide,  3J  inches  thick, 
and  20,  16,  17,  19,  23,  10,  7,  12,  6,  and  20  feet 
long?  The  sum  of  the  lengths  is  150  feet: 
hence,  we  place  it  on  the  right,  thus, 


2—, 


* 

4 


150 

—3 
15 


8|3375 


j—  3 


Here,  40  planks  in  a  pile,  is 
placed  down  once  for  the  whole 
lot,  considering  that  the  lot  is 
now  150  feet  long.  The  answer 
is  in  dollars,  because  the  price 
was  dollars. 

What  will  80  pieces  of  lumber, 
8  feet  long,  9  inches  wide,  and 
2i  inches  thick,  come  to,  at  60 
cents  per  hundred  ? 

We  deem  the  examples  given, 
sufficient  for  the   measurement 
of  lumber,  as  there  is  but  very 
little  difficulty  in  the  statement. 

How  many  cubic  feet  in   a  stick  of  timber 
30  feet  long,  8  inches  thick,  and  10  inches  wide  ? 
In  this  example  it  is   ne- 
cessary to    divide    both    the 
width  and  the  thickness  by 
12,  to  reduce  them  to  feet, 
that  by  multiplying  all  the  di- 
mensions in  feet,  the  product 
may  be  solid  feet.     Hence, 


60 


jlGfft. 


To  measure  lumber,  Place  the  length  in  feet, 
the  width  in  inches,  and  the  thickness,  in  incJics, 
on  the  right ,  and  12  on  the  left. 

To  ascertain  the  number  of  feet  in  the  whole 
pile,  when  of  the  same  dimensions,  place  the  number 


MASONRY.  223 

of  picas,  likewise  on  tlw  right.  If  the  answer  is 
desired  in  dollars,  or  dollars  and  cents,  place  100 
on  the  left,  and  the  price  per  hundred,  on  the  right. 
To  lose  a  fraction  for  saw-cut,  subtract  the 
fraction  lost  from  such  a  number  of  parts  of  the 
same  she  as  would  constitute  a  unit,  place  the 
remainder  on  the  right,  and  the  number  making  a 
unit,  on  the  left. 

MASONRY. 

Masonry,  as  a  department  of  measurements, 
may  properly  be  classed  with  cubic  timber 
measure.  Stone  work  is  measured  by  the 
perch,  which  is  generally  25  solid  feet,  or  16  J 
feet  long,  \\  feet  wide,  and  1  foot  high.  A 
solid  perch  in  masonry,  is  a  mass  162  feet  in 
every  way.  The  word  perch  is  derived  from 
the  French  perche,  which  signifies  sharp,  extend- 
ing, fyc.,  as  a  pole  or  rod  for  measurements. 
Hence  the  name  is  derived  from  the  limits 
which  define  it. 

How  many  perches  of  masonry  in  a  wall  80 
feet  long,  15  feet  high,  and  2i  feet  thick? 

Here,  we  make    25    solid  ,   , £k\$Q 4 

feet    a   perch,    saying,  what 
will  all  of  the  feet  in  the  wall 


— 3 


make,  if  25  opposite  make  1  — r^r 

perch.     Hence  120  perches. 

We  may  easily  ascertain  the  price  for  a 
piece  of  work  at  the  same  time  that  the  quan- 
tity is  obtained,  by  placing  the  price  per  perch 
last  on  the  right. 


224  RAINEY'S  IMPROVED  ABACUS. 

^o ~  What  will  it  cost  to  put  up 

a  wall  200  feet  long,  6  feet  3 
inches  high,  and  3f  ft.  thick, 
at  120  cents  per  perch  of  25 


15 


The  price  being  in  cents,  2 
figures  are  cut  off  at  the  right  of  the  answer, 
for  cents. 

How  much  will  it  cost  to  wall  a  cellar,  at 
$1,60  cents  per  perch,  20  feet  square,  and  7i 
feet  deep,  with  a  wall  li  feet  thick? 

/ £,  ££ 27  It  is  evident  that  the  two 

end  walls  are  each  3  feet 
shorter  than  those  of  the 
sides  :  hence,  the  entire  length 


— 4 


|1,60 


^h  I  CO     QO 

place  this,  with  the  height 
and  thickness,  on  the  right,  and  the  denomi- 
nators on  the  left.  We  use  the  factor  5  on  the 
two  sides  of  the  line.  Hence, 

To  ascertain  the  number  of  perches  in  a  piece 
of  stone  work,  place  the  length,  height,  and  width, 
in  feet,  on  the  right,  and  25,  or  whatsoever  stand- 
ard is  acknowledged,  on  the  left:  the  answer  will 
be  solid  perches.  If  the  cost  is  desired,  place  the 
price  per  perch  last  on  the  right,  and  the  answer 
will  be  tJie  cost  of  the  entire  work,  in  dollars  or 
cents. 

PLASTERERS',  PAVERS',  AND  BUILDERS'  WORK. 

Plasterers  and  Pavers  calculate  their  work 
by  the  square  yard,  or  9  square  feet :  Builders 
reckon  by  the  square,  which  is  100  square 


PLASTERERS'  AND  PAVERS'  WORK. 


225 


110 


1 80  yards. 


0 

18 
4£—  2 

36 
80 

feet,   in    weather-boarding,   ceiling,   framing, 
shingling,  &c. 

How  many  square  yards  of  plastering  in  a 
room  18  feet  square,  and  10  feet  high.  We 
place  the  side,  18,  on  the  right,  and  4,  which 
will  give  all  the  sides. 

Having  the  dimensions  in 
feet  on  the  right,  we  place  9 
feet,  which  make  a  square 
yard,  on  the  left :  the  answer 
is  the  sum  of  the  four  sides, 
80  yards.  We  now  place  18 
on  the  right  of  another  line 
twice,  and  ascertain  the  num- 
ber of  yards  overhead,  by  the 
same  process,  which  is  36. 
The  two  added,  make  the 
number  of  yards  in  the  room,  116.  We  might 
have  ascertained  the  cost  of  the  whole,  quite 
as  easily,  by  placing  the  price  in  each  state- 
ment, last  on  the  right. 

What  will  the  plastering  of  a  room  come  to, 
which  is  15  by  20  feet,  and  12  feet  high,  at 
22i  cents  per  yard? 

The  4  sides  make  70  feet 
around,  which  we  place  with 
the  height  and  price,  on  the 
right.  The  cost  of  the  sides  is 
$21,00.  Again,  we  place  15 
and  20  on  the  right,  with  the 
price,  and  9  on  the  left.  This 
makes  the  plastering  over- 
head come  to  7  dollars  and 
50  cents,  which,  added  to 
the  sum  above,  makes  the 


70 


—  2 


21,00 
7,50 


$|28,50 


t— 5 


$|7,50 


226 


RAINEY'S  IMPROVED  ABACUS. 


300—25 


75 


$|56,25 


cost  of  the  room  amount  to  28   dollars  and 
50  cents. 

How  much  will  it  cost  to  lay  a 
pavement  300  feet  long,  and  4 
feet  6  inches  wide,  at  37i  cents 
per  square  yard  ? 

This  question  is  identical  with 
those  just  wrought  in  plastering.     Hence, 

To  ascertain  the  number  of  yards  of  plastering,  or 
paving,  place  the  whole  length  of  the  walls,  or  pave, 
with  the  width  or  height  in  feet,  on  the  right,  and 
9  on  tJie  left:  if  the  answer  is  wished  in  money, 
place  the  price  per  square  yard,  last  on  the  right : 
the  answer  will  be  the  price  of  the  whole. 

How  many  squares  of  weather-boarding  on 
a  building  50  by  40  feet,  21  feet  high?  What 
will  the  same  come  to,  at  $1,50  per  square? 

The  whole  length  of  the  build- 
ing, or  sum  of  the  sides,  is  placed 
on  the  right,  with  the  price,  and 
i  100,    the    number    of    feet    in    a 

«JP    4O,    /U         I  .-l  -i         n 

1  square,  on  the  left. 

What  does  it  cost  to  shingle  a  roof  80  feet 
long,  and  20  feet  from  the  eaves  to  the  cone,  at 
87i  cents  per  square  ? 

The  roof  is  80  by  40  feet,  which 
dimensions  are  placed  on  the 
right,  with  the  price.  Thus,  by 
proportion,  what  will  all  of  these 
feet  come  to,  on  the  right,  if  100 
feet,  opposite,  cost  87i  cents?  The  answer 
is  $28.  Hence, 

To  ascertain  the  cost  of  weather-boarding,  shingling, 
framing,  flooring,  <fyc.,  place  the  length  and  width  in  feet,  on 
the  right ;  ]  00  on  tlie  left ;  and  the  price  per  square,  last  on 
the  right. 


180 

21 

1,50 


m 


40-2 
176 


CORN,  COAL,  AND  LIQUID  MEASURE.  227 


CRIBS,    BOXES,    AND    BODIES. 

The  standard  of  measurements  of  this  kind,  is  gener- 
ally inches;  the  number  of  inches  making  a  bushel,  a 
gallon,  &c. 

A  compact  bushel,  as  wheat,  shelled  corn,  salt,  <fyc.9  contains 
2150i  cubic  inches,  which  may  be  expressed  and  used  deci- 
mally, in  the  form  of  2150,2. 

A  dry  busJiel,  as  potatoes,  apples,  coal,  <$-c.,  contains  2688 
cubic  inches. 

A  wine  gallon  contains  23 1 :  a  beer  gallon,  282  cubic  inches. 

A  solid  foot  contains  1728  cubic  inches. 

If  the  length,  width,  and  height  of  a  body,  crib,  or  box, 
are  placed  on  the  right  of  the  line.,  their  product  will  be 
the  number  of  cubic  inches  in  such  body,  crib,  &c.  The 
question  then  is,  by  proportion,  what  will  all  these  inches 
on  the  right  give,  if  2150  opposite,  give  1  bushel ;  or,  if 
2688  give  one  dry  bushel;  or,  231  give  1  wine  gallon;  or, 
if  282  give  1  beer  gallon,  &c.  1  Hence,  after  these 
dimensions  are  placed  on  the  right,  it  is  only  necessary 
to  place  the  number  making  a  unit  of  the  desired 
measure,  on  the  left.  If,  in  measuring  *coal,  the  body  is 
wider  at  the  top  than  at  the  bottom,  take  the  mean,  width, 
by  measuring  half  way  between  the  top  and  bottom. 

The  2150  inches  used,  make  an  even  bushel;  hence, 
when  the  corn  is  in  the  ear,  place  10  on  the  left :  when 
in  the  husk,  place  20  on  the  left,  and  the  answer  will  be 
in  barrels.  This  allows  5  bushels  of  shelled  corn,  10  of 
unshelled,  and  20  in  the  husk,  for  a  barrel. 

When  it  is  necessary  to  ascertain  the  whole  price,  the 
price  per  bushel,  &c.,  may  be  placed  last  on  the  right. 


160 


How  many  bushels  corn  in  a  crib 
160  inches  long,  86  inches  wide,  and 
90  inches  high  1 

TKe  factor  5  is  hero  used  in  the 
10,  and  215. 

How  many  bushels  in  a  crib  120 
inches  long,  45  wide,  and  80  high  "? 

We  frequently  find  it  impracticable  I                    |80 
to  cancel  in  these  calculations.    This,  !"                  [8640~ 
however,  is  of  but  little  moment.         j -- — 


228 


RAINEY'S  IMPROVED   ABACUS. 


100 

£0 


$|100,00 


100-15 
W 


$|4,50 


1 1000  galls/ 


47 


What  will  a  crib  of  corn  come 
to,  at  21^  cents  per  bushel,  200 
inches  long,  100  wide,  and  50 
deep] 

This  combination  of  state- 
ment is  quite  simple  and  easy. 

What  will  a  load  of  coal  cost, 
at  three  and  a  half  cents  per 
bushel,  which  measures  120  in. 
long,  48  inches  wide,  and  60 
inches  high  ? 

The  price,  4  dollars  and  50 
cents,  pays  for  the  whole  load. 

How  many  gallons  water  in  a 
tan  vat,  70  inches  long,  60  inches 
wide,  and  55  inches  deep  ? 

The  answer  is  1000  gallons,  in 
measure. 

How  much  will  a  vat  of  beer 
come  to,  at  50  cents  per  gallon, 
the  vat  being  120  inches  long, 
40  inches  deep,  and  20  inches 
wide  ! 

The  answer  is  170  dollars,  21 
170,211^  cents,  and  a  fraction. 

To  find  the  contents  of  a  crib,  body,  or  box,  in  compact 
bushels,  place  all  the  dimensions  on  the  right,  in  inches,  and 
on  the  left  2150.  If  the  answer  is  wished  in  barrels,  place 
5  on  the  left  for  shelled  corn,  10  for  corn  in  the  ear,  and  20 
for  corn  in  the  husk. 

When  the  answer  is  desired  in  dry  bushels,  place  2688, 
instead  of  2 1 50,  on  the  left.  In  either  case,  place  the  price,  on 
the  right,  last,  and  the  answer  will  be  the  price  of  the  whole. 

To  ascertain  the  number  of  gallons,  place  the  dimensions, 
as  above,  on  the  right,  and  for  wine  gallons  231,  or  for  beer 
gallons  282,  on  the  left.  The  price  per  gallon  may  be  placed 
last  on  the  right ;  the  answer  will  be  the  price  of  the  whole. 

When  the  contents  of  any  crib,  box,  or  body,  and  two  of 
the  sides  are  given  to  ascertain  the  other  side,  place  the  con- 
tents and  the  standard  of  unity,  or  the  number  of  inches  which 
make  a  unit  of  the  contents,  on  the  right,  and  the  two  given 
dimensions  on  the  left :  the  answer  will  be  the  required  side. 


40—2 


50 


1 800000 


TONNAGE    OF    VESSELS. 


229 


A  body  is  120  inches  long,  and  86  inches  wide  ; 
how  high  must  it  be  to  hold  180  bushels  of  coalV 

It  is   seen  here,   that   the     2—  £#0  #0^—224 
body  must  be  112  in.  high;  or       #  —  #0  j/L$0  —  £p  —  # 

9  ft.  4  in.     Again,  ~  r~^     ~ 

112  inches. 

A  crib  is  215  inches  long,  and  100  inches  wide  ; 
how  high  must  it  be  to  hold  1500  bushels  of  corn? 


Thus,   the   crib  must    be    150  in., 
equal  to  12|  feet  high. 


1500 


150  in. 


TONNAGE   OF    VESSELS. 

In  ascertaining  tonnage,  it  is  necessary  to  ascertain 
as  nearly  as  practicable,  the  number  of  cubic  feet  of 
water  displaced  by  the  vessel.  This  is  done  by  mul- 
tiplying together  the  length,  width,  and  depth  of  the 
vessel,  which  gives  the  number  of  cubic  feet  con- 
tained in  the  hull.  Now,  it  is  a  law  of  hydrostatics, 
that  "if  a  body  floats  on  a  fluid,  it  displaces  as  much, 
of  the  fluid,  as  is  equal  to  its  own  weight"  Nor 
does  it  make  any  difference  what  the  shape  of  such 
body  be ;  a  quantity  of  water  equal  to  its  own  weight 
must  be  displaced.  Hence,  in  ascertaining  the  weight 
that  any  hollow  square  will  sustain  in  water,  it  is  ne- 
cessary, first,  to  ascertain  the  weight  of  water  such 
square  would  contain;  making  all  due  allowance  for 
weight  of  vessel,  room  for  safety,  etc.,  etc.  A  cubic 
foot  of  water  weighs  1000  ounces  avoirdupois,*  or 
62i  Ibs.;  hence,  95,  the  number  of  cubic  feet  allowed 

*The  cubic  foot  of  distilled  water  weighs  about  1000  oz. 
avoirdupois,  or  very  nearly  62J£  Ibs.,  at  40°  temperature;  at 
GO0,  which  is  generally  used,  It  weighs  only  62,353  Ibs.,,  less 
than  1000  oz.  The  foot  weighs  911.458  oz.  troy,  or  .5274  oz 
per  cubic  inch.  The  cubical  foot  equals  2200  cylindrical, 
3300  spherical,  or  (F600  conical  inches.  A  cylindrical  foot  of 


230  RAINEY'S  IMPROVED  ABACUS. 

for  1  ton,  or  2240  Ibs.  of  freight,  will  weigh  5937|lbs. 
avoirdupois.  This  allows  nearly  three  tiroes  the 
weight  of  the  freight  in  water,  to  the  ton. 

There  are  two  methods  used  in  reckoning  tonnage, 
the  carpenters',  and  the  government  rule. 

The  following  is  the  government  rule  for  measuring 
single  or  double-decked  vessels : 

"  If  the  vessel  be  double-decked,  take  the  length  thereof,  from  the  fore 
part  of  the  main  stem,  to  the  after  part  of  the  stern  post,  above  the  upper 
deck;  the  breadth  thereof  at  the  broadest  part  above  the  main  wales,  half 
of  which  breadth  shall  be  accounted  the  depth  of  such  vessel;  and  then 
deduct  from  the  length  three-fifths  of  the  breadth,  multiply  the  remainder 
by  the  breadth,  and  the  product  by  the  depth,  and  divide  thi's  product  by 

water  weighs  49.1  Ibs.,  avoirdupois;  a  cylindrical  inch,  .02642, 
and  a  cubic  inch,  .03617  Ibs.,  avoirdupois. 

Seawater  weighs  1.03  times  distilled  water,  which  is  the 
standard  of  weight:  hence,  1  cubic  foot  of  seawater  weighs 
1030  oz.,  avoirdupois.  f 

Nineteen  cubic  inches  distilled  water,  temperature  50° 
Fahr.,  weigh  10  oz.  troy,  according  to  act  of  parliament,  1825. 

It  may  be  observed  here,  that  the  standard  of  liquid  measure 
MI  the  United  States,  is  the  wine  gallon,  containing  231  inches, 
equal  to  8.339  Ibs  avoirdupois,  or  58372.1754  grains  distilled 
water. 

The  English  imperial  standard  gallon  is  10  Ibs.,  avoirdupois, 
distilled  water,  at  62°  Fahr.,  and  30  inches  barometer:  and  is 
about  equal  to  277.274  inches.  It  is  about  equal  to  one 
and  one-fifth,  or  1.2  gall.,  wine  measure  of  the  U.  States. 

In  Great  Britain,  the  imperial  bushel  weighs  80  Ibs.,  avoirdu- 
pois, distilled  water,  at  62°  Fahr.,  and  30  in.  of  the  barometer. 
It  is  a  vertical  cylinder,  18,789  inches  in  diameter,  and  8  inches 
deep;  and  contains  2218.192  cubic  inches.  This  standard  is 
adopted  in  New  York. 

The  United  States  standard  of  dry  measure  is  the  Winches- 
ter bushel,  containing  77.627413  Ibs.,  avoirdupois,  distilled 
water,  maximum  density,  and  weighed  in  air  at  30  in.  barom- 
eter. It  contains  2150.42  cubic  inches  nearly,  although 
2150.2  is  more  used. 

In  Connecticut,  2198  cubic  inches  make  a  bushel.  The 
measure  used,  varies  in  the  different  states:  hence,  the  pro- 
priety of  knowing  the  standard  of  the  state  in  which  the  cal- 
culation is  made.  It  should  be  the  same  in  all  of  the  states, 
as  is  our  currency. 


TONNAGE    OF    VESSELS. 


231 


95,  the  quotient  whereof  shall  be  deemed  the  true  contents  or  tonnage 
of  such  ship  or  vessel:  and  if  such  ship  or  vessel  be  single-decked,  take 
the  length  and  breadth,  as  above  directed,  deduct  from  the  said  length 
three-fifths  of  the  breadth,  and  take  the  depth  from  the  under  side  of  the 
deck  plank,  to  the  ceiling  of  the  hold;  then  multiply  and  divide  as  afore- 
said, and  the  quotient  shall  be  deemed  the  tonnage." 

The  foregoing  rule  may  be  used  to  get  the  dimen- 
sions ;  after  this,  the  dimensions  may  be  placed  on  the 
right  of  the  line,  to  find  the  entire  number  of  feet  in 
the  boat,  and  95  on  the  left.  The  proportion  will  be, 
as  95  to  the  continued  product  of  these  dimensions,  so 
will  1  ton,  which  the  95  feet  equals,  be  to  the  whole 
number  of  tons  in  the  vessel. 

What  is  the  government  tonnage  of  a  single-decked 
vessel,  115  ft.  keel,  25  ft.  beam,  and  10  ft.  hold?  * 

Here,  f  of  the  width,  which  is  25  feet,  equals  15 
feet,  to  be  subtracted  from  the  length.  This  leaves 
the  length  100  feet.  The  dimensions  are  placed  on 
the  right,  and  95  on  the  left ;  thus, 


In  this  instance,  the  answer  is 
263 T3g-  tons.  In  carpenters'  mea- 
sure, the  |  of  the  width  would  not  be 
taken  off,  but  would  be  calculated, 
thus, 


Here,  the  answer  is  302|f  tons. 


100 


10 


5000 


9__0£  115 


10 


5750 


302|f 

How  many  tons  in  a  double-decked  vessel,  268  ft. 
long,  30  ft.  wide,  and  15  ft.  deep  ? 

Here  J  of  the  width,  18ft.,  are  subtracted  from  the 

*The  keel  of  a  vessel  is  the  main  bottom  in  length:  the  beam  is  the 
greatest  width  from  side  to  side  of  the  hull;  and  the  hold,  the  'depth  from 
the  main  deck  to  the  bottom  of  the  hull. 


232 


RAlffEY  g   IMPROVED    ABACUS. 


length,  leaving  250  ft.  long.     The  depth  being  ^  of 
the  width,  is  15  ft.:  we  state  accordingly, 
:250 
30 

From  this  calculation,  the  ton- 
nage is  1184T4¥. 


22500 


What  is  the  tonnage,  according  to  carpenters'  mea- 
sure, of  a  steamboat,  300  ft.  keel,  10ft.  hold,  and  38 
ft.  beam? 

We  here  place  all  of  the  dimensions  on  the  right, 
and  divide  by  95 ;  thus, 

300  The  factor,  5,  is  contained  in  10 

twice,  and  in  95,  nineteen  times :  19 
into  38,  twice,  and  2x2x300= 
1200  tons,  the  answer.  Hence, 


Tons. 


1200 


TO    ASCERTAIN    TONNAGE. 

Ascertain  frst  the  dimensions  of  the  vessel,  according  to  gov- 
ernment, or  carpenters'  rule:  place  the  dimensions  on  the  right 
of  the  line,  and  95  on  the  left:  the  answer  will  be  the  number  of 
tons. 

To  ascertain  any  given  dimension,  when  two  of  the  dimen- 
sions and  the  tonnage  are  given,  place  the  tonnage  and  95  on  the 
right,  and  the  two  given  dimensions  in  feet,  on  the  left:  the  an- 
swer will  be  the  required  dimension  in  feet. 


SUPERFICIAL   GEOMETRY. 

VARIETY    OF    FIGURES,    DEFINITIONS,    ETC.,    ETO. 

A  line  is  the  shortest  possible  distance  between  two 
points.  A  line  is  supposed  to  be  straight;  when 
curved,  the  word,  curved,  is  mentioned. 

Two  straight  lines  are  parallel,  when  equally  dis- 


DEFINITION    OF    TERMS    IN      GEOMETRY.          233 

tant  at  all  parts ;  and  when  they  cannot  come  togeth- 
er, however  far  extended  or  produced.* 

The  point  of  intersection  between  two  lines,  is 
called  an  angle. ,f 

An  angle  of  90  degrees,  or  ^  of  the  circumference 
of  a  circle,  is  called  a  right  angle :  an  angle  of  less 
than  90  degrees,  is  called  an  acute,  or  sharp  angle. 

An  angle  of  more  than  90  degrees,  is  called  an  ob- 
tuse, or  blount  angle. 

A  figure  with  four  equal  angles,  and  four  equal 
sides,  is  called  a  square.^. 

A  figure  with  four  equal  angles,  and  unequal  sides, 
is  called  a  rectangle.\\ 

A  figure  with  four  parallel  unequal  sides,  and  two 
acute,  and  two  obtuse  angles,  is  called  a  parallelo- 
gram^ or  rhomboid.** 

A  figure  with  four  equal  sides,  and  two  obtuse  and 
two  acute  angles,  is  called  a  rhomboid;  sometimes,  a 
lozenge. ,ft 


*  The  word  produced,  in  this  instance,  bears  its  literal  signi- 
fication, lead  out,  from  the  Latin,  pro  and  duco. 

t  Angle  is  a  French  word,  which  is  from  the  Latin,  angu- 
lus,  a  corner. 

J  Square  is  derived  from  the  French,  quarre,  which  is  ori- 
ginally from  the  Latin,  quatuor,four. 

||  Rectangle  is  from  the  Latin,  rectus,  right,  and  angulus,  an 
angle. 

§.Parallelogram  is  derived  from  the  Greek,  9r«t/>atXA»xof,  or 
parallelos,  opposite  the  one  to  the  other,  and  y^ppa.,  or  gram- 
ma, a  character  or  figure. 

**  Rhomboid  is  from  the  Greek,  /Joyu/So?,  or  rorribos,  a  rhomb, 
and  wJoff,  or  eidos,  form.  Rhomb  is  from  the  Latin  rhombus, 
a  whirl,  or  constantly  varying  square  :  in  fabulous  history,  a 
varying  or  rolling  instrument  by  which  witches  were  said  to 
bring  down  the  moon  from  heaven. 

ft  Lozenge  is  from  the  Gr.  AO£OC  or  loxos,  oblique,,  and  yuvk*. 
or  goniay  a  corner 


234  RAINEY'S  IMPROVED  ABACUS. 

A  figure  with  four  unequal  angles,  and  two  parallel 
sides,  is  called  a  trapezoid* 

A  figure  having  three  or  more  equal  sides,  is  called 
a  polygon.^ 

The  lowest  polygon,  that  of  three  sides,  is  called  a 
triangle ;J  that  of  four  sides,  a  quadrilateral;  of  five 
sides,  a  pentagon;  of  six  sides,  a  hexagon;  of  seven 
sides,  a  heptagon;  of  eight  sides,  an  octagon;  of 
nine,  a  nonagon;  of  ten,  a  decagon;  of  eleven,  an 
undecagon ;  and  of  twelve,  a  dodecagon. 

There  are  four  principal  triangles  :  the  equilateral, 
the  isosceles,  the  scaline,  and  the  rectangle  triangle. 

A  triangle  which  has  three  sides  is  called  equilate« 
ral  :\\  one  which  has  two  of  its  sides  equal,  is  called 
isosceles  :§  one  which  has  three  unequal  sides,  is  called 
scaline  :^[  and  that  which  has  a  right  angle,  is  called  a 
rigM  angled,  or  rectangle  triangle. 

The  longer  arm  of  the  right-angled  triangle  is  called 
the  basefoom  the  Lat.  basis,  the  bottom  or  foundation  : 
the  shorter  arm  is  called  the  side  ;  and  the  side  opposite 
the  right  angle,  the  hypotenuse,  from  the  participle  of 

*  Trapezoid  is  from  the  Gr.  rpstTrt^tov,  or  trapezion,  a  small 
table,  and  s^To?,  or  eidos,form. 

f  Polygon  is  from  the  Gr.  TTOMZ,  many,  and  ywta,  an  angle. 

t  Triangle  is  from  the  Lat.  triangulum,  from  tres,  three,  and 
angulus,  a  corner:  hence,  a  figure  with  three  angles. 

Quadrilateral  is  from  the  Lat.  quatnor,four,  and  latus,  side; 
having  four  sides  :  pentagon,  from  the  Gr.  WSVTS,  or  pente, 
jive,  and  yu>vi±,  or  gonia,  a  corner.  Hexegon,  heptagon,  octo- 
gon,  nonagon,  decagon,  undecagon,  and  dodecagon  are  com- 
pounded by  prefixing  to  the  yavtx,  or  gonia,  t%,  or  ex,  six;  sirra., 
or  epta,  seven;  •x.ra),  or  okto,  eight;  Lat.  nonus,  nine;  JMA,  or 
deca,  ten;  Lat.  undecim,  eleven;  JWSKA,  or  dodeka,  twelve,  etc. 

||  Equilateral  is  derived  from  the  Lat.  aquus,  equal,  and  late- 
ralis,  from  latus,  a  side:  equal  sided. 

§  Isosceles  is  derived  from  the  Gr.  ires  or  isosf  equal,  and 
CTMSXO?  or  skelos,  a  leg:  hence  it  has  two  equal  legs. 

If  Scaline  is  from  the  Gr.  O-X.X.XMK,  or  skalenos,  that  totters  or 
hangs  over  to  one  side,  obliquely. 


GEOMETRICAL    FIGURES.  235 

the  Greek  verb  wroTwovo-et9  or  upoteinousa,  extending 
under,  or  from  corner  to  corner. 

Any  of  the  sides  of  an  equilateral  and  scaline  tri- 
angle may  be  called  the  base.  The  base  of  an  isos- 
celes triangle  is  the  short  side. 

The  upper  point  where  the  two  sides  of  an  isosce- 
les, or  other  triangle,  meet,  is  called  the  vertex  of  the 
triangle;  by  some,  the  apex.* 

The  theory  of  determining  angles,  and  the  lengths 
of  the  sides  of  triangles,  constitutes  the  science  of 
Trigonometry ;  and  cannot  be  properly  treated  in  arith- 
metic. We  have  before  seen  that, 

To  find  the  contents  of  a  square  or  rectangle,  multiply  the 
two  sides  together. 

A  parallelogram  is  equal  in  contents  to  a  square  or 
rectangle,  when  its  base  and  verticalf  hight  are  equal 
to  the  base  and  side  of  such  square  or  rectangle 

A  rhomboid  or  lozenge  is  equal  to  a  square  of  the 
game  base  and  altitude.  Hence, 

To  find  the  contents  of  a  parallelogram,  or  rhomboid,  multi- 
ply the  base  by  the  vertical  hight. 

To  find  the  contents  of  a  trapezoid,  multiply  half  the  sum  of 
the  two  parallel  sides,  by  the  vertical  distance  between  them. 

How  many  yards  of  plastering  in  a  wall  30  feet 
square  ? 

We  divide  by  9  feet,  which  make  1  square 
yard.  100 

*  These  two  words  are  frequently  used  synonymously. 
Some  apply  apex  to  triangles,  and  vertex  to  cones,  because  of 
the  primary  signification  of  vertex,  from  the  Lat.  vertex,  a 
point,  which  is  from  verto,  to  turn.  The  plural  of  vertex,  is 
vertices;  and  of  apex,  apices. 

fThe  vertical  hight  of  any  quadrilateral  or  four-sided 
figure,  is  a  line  dropped  from  the  upper  plane  or  vertex,  per- 
pendicular to  the  base.  Hence,  the  side  of  a  parallelogram 
must  not  be  multiplied  into  the  base  for  the  contents,  as  thte 
is  too  long,  but  the  side  arising  from  a  line  dropped  at  right 
angles  with  the  base. 


236 


RAINEY'S  IMPROVED  ABACUS. 


How  many  acres  are  there  in  the  road  from  Cincin- 
nati to  Dayton,  which  is  64  miles  long,  and  4  rods 
wide? 


64 


Ans. 


512  A. 


Here,  we  ask  how  many  rods  64 
miles  will  make,  if  1  mile  make  8 
furlongs,  and.  1  furlong  make  40 
rods ;  then,  multiplying  by  4  rods  in 
width,  we  say,  how  many  acres  will 
all  these  rods  make,  if  160  rods 
make  1  acre?  Ans.  512  acres. 


The  first  question  relates  to  a  square,  or  equal 
rhomboid;  the  second,  to  a  rectangle,  or  equal  paral- 
lelogram. 

The  two  parallel  sides  of  a  trapezoid  are  40  and  60 
rods,  and  the  distance  between  them  80  rods :  how 
many  acres  are  there  ? 
i  £0—25 
#0  The  answer  is  25  acres. 


Ans.  25  acres. 


TRIANGLES. 


Every  triangle  is  half  of  a  square,  rectangle,  paral- 
lelogram, or  rhomboid  of  similar  base  and  altitude. 
To  prove  this,  let  us,  on  the  hypotenuse,  or  longest 
side,  of  any  given  triangle,  erect  another  triangle,  with 
the  side  and  base  parallel  and  equal,  each,  to  the  side 
and  base  of  the  triangle.  The  figure  formed  will  be 
a  square,  rectangle,  parallelogram,  or  rhomboid,  which 
proves  the  position  correct.  Hence, 

To  find  the  contents  of  any  triangle,  multiply  half 
of  the  base  by  the  whole  vertical  hight,  or  the  whole 
base  by  half  of  the  vertical  hight,  as  may  be  most 
convenient. 

How  many  acres  of  land  in  a  rectangle  triangle,  of 
240  rods  base,  and  120  rods  side  ? 


THEORY    OF    TRIANGLES. 


237 


Ans. 


90  A. 


Here,  we  place  down  the  whole  '  £ — fi.$0  fififi — 3 
base   and   side,  and   divide   by  2,  #  1#0 — 3 

which   is   both   easy   and   simple ; 
while,  likewise,  we  have  the  advan- 
tage of  dividing  by  such  denominate  numbers  as  are 
necessary  to  reduce  to  a  given  denomination. 

How  many  square   feet  in  an   isosceles  triangle, 
of  16f  ft.  base,  and  37^  ft.  vertical  hight? 

50—5 

c\ 

Here,  2  is  thrown  on  the  left;  and  we 
divide  by  the  denominators,  3  and  2. 
Hence,  the  answer,  312^  square  feet. 


625 


If  a  line  dropped  from  the  vertex  of  a  scaline  tri- 
angle, fall  outside  of  the  triangle,  the  line  of  the  base 
must  be  produced  until  it  meets  the  vertical  line. 

The  square*  of  the  hypotenuse  of  a  right-angled 
triangle,  is  equal  to  the  sum  of  the  squares  of  the 
base  and  side. 

For  example  ;  the  base  of  a  right-angled  triangle  is 
8,  the  side  6,  and  the  hypotenuse  10.  Now,  8x8= 
64;  and  6><6=36;  and  36+64=100;  hence,  the 
hypotenuse  is  10x10=100,  which  is  the  sum  of  the 
squares  of  the  base  and  side.  A  rectangle,  whose 
sides  are  4,  3,  and,  5,  shows  the  same  equality;  thus, 
4X4=16;  3x3=9;  and  16+9=25:  now,  5x5 
=25,  which  proves  again  that  the  sum  of  the  squares 
of  the  two  sides  equals  the  square  of  the  hypotenuse. 

Frem  the  foregoing,  it  follows,  that 

To  find  the  hypotenuse  of  a  right-angled  triangle^ 

*The  square  of  any  number,  is  that  number  multiplied  into  itself;  49  is 
tbe  square  of  7-  When  it  is  desired  to  square  a  number,  2  is  written  over 
it,  thu§,  72 

1G 


288  RAINEY'S  IMPROVED  ABACUS. 

extract  the  square  root  of  the  sum  of  the  squares  of 
the  base  and  side.* 

To  find  the  base,  when  the  hypotenuse  and  side  are 
given,  subtract  the  square  of  the  side  from  the  square 
of  the  hypotenuse  ;  extract  the  square  root  of  the  re- 
mainder, and  the  answer  will  be  the  base. 

To  find  the  side,  when  the  hypotenuse  and  base  are 
given,  subtract  the  square  of  the  base  from  the  square 
of  the  hypotenuse  ;  extract  the  square  root  of  the  re- 
mainder,  and  the  answer  will  be  the  required  side. 

The  hypotenuse  of  a  right-angled  triangle  is  10  ft., 
and  the  side  6  ft.;  what  is  the  base  ? 

The  square  of  the  hypotenuse  is  10x10=100; 
and  of  the  side  6x6=36:  now,  100—36=64,  and 
the  square  root  of  64,  is  8,  which  is  the  required  base. 

POLYGONS. 

If  lines  be  drawn  from  the  angles  of  the  polygon, 
to  the  center,  it  is  manifest  that  the  polygon  will  be 
divided  into  isosceles  triangles ;  and  if  the  contents 
of  one  of  these  be  multiplied  by  the  whole  number  of 
angles  thus  made,  the  answer  will  be  the  contents  of 
the  polygon.  Hence, 

To  find  the  contents  of  a  regular  polygon,  multiply 
half  of  the  diameter  of  the  polygon,  vertical  to  one 
of  the  sides,  by  half  of  one  of  the  sides;  and  the 
product  by  the  number  of  sides. 

Or,  Place  the  shortest  semi-diameter*,  and  one  oj 
the  sides,  on  the  right  of  the  line,  and  2  on  the  left. 

*  Carpenters  frequently  use  this  method  of  finding  the  length  of  braces, 
where  the  distance  from  the  angle,  or  lower  end  of  the  post,  to  the  extreme 
end  of  the  mortice,  is  given. 

W!,en  the  base  and  side  of  a  rectangle  triangle  are  of  the  same  length, 
th*1  hypotenuse  may  be  found  by  multiplying  the  base  or  side  by  1.4142. 
This  number  is  the  square  root  of  twice  3J41592,  which  is  the  ratio  of  the 
circumference  to  the  circle. 

t  Semi-diameter  means  half -diameter,  from  the  Latin  semi,  half.  Ra- 
dius is  used  to  denote  the  same  thing  in  circles. 


SPHERICAL   MEASUREMENTS.  239 

THE    CIRCLE.* 

Squares  are  always  used  as  the  units  of  superficial 
measurement,  by  reason  of  their  sides  and  angles  co- 
inciding, and  leaving  no  intervening  space  between 
their  limits.  The  unit  assumed  is  generally  a  square 
inch,  a  square  foot,  a  square  yard,  a  square  mile,  etc. 

A  circle  is  a  plain  figure,  bounded  by  a  line  called 
the  circumference,^  which  is,  at  all  parts,  equally  dis- 
tant from  a  point  within,  called  the  centre ;  hence, 

The  circumference  of  a  circle  is  a  line  drawn  at  all 
parts  equally  distant  from  the  centre. 

The  diameter  %  of  a  circle  is  a  straight  line  drawn 
from  the  opposite  sides  of  the  circumference,  through 
the  centre,  dividing  the  circle  into  two  equal  parts. 

The  periphery  \\  of  a  circle  is  its  circumference;  the 
two  words  being  used  synonymously,  at  pleasure. 

The  radius^  of  a  circle  is  a  line  drawn  from  the 

*  Circle  is  derived  from  the  Latin  circus,  a  round  ring,  or 
limit;  or  from  the  Gr.  M/MOS  or  kirkos,  a  falcon,  that,  in  flying, 
describes  circles  ;  or  from  the  Arabic  kara,  to  go  round. 
Many  individuals  confound  circle  with  circumference;  where- 
as, while  the  latter  merely  describes  the  limits,  the  former  is 
the  space  included  in  such  limits. 

t  Circumference  is  from  the  Lat.  circum,  around,  and/ere/i- 
tia,  from  fero,  to  bear. 

Centre  is  a  French  word,  from  a  Gr.  noun,  signifying  a 
goad  or  point,  which  is  from  the  root  KWWM  or  kenteo,  to  prick. 

t  Diameter  is  from  the  Gr.  Jin  or  dia,  through,  or  through  the 
middle,  and  jutrpsv  or  mttron,  to  measure. 

||  Periphery  is  from  iryt  or  peri,  around,  about,  and  pg/>a>  or 
•fthero,  to  bear:  hence,  it  is  identical  with  the  circumference. 
The  word  perimeter  is  sometimes  used  in  the  same  sense,  but 
improperly.  It  relates  particularly  to  the  measurement  or  ex- 
tent of  circumferences,  from  peri  and  metron,  to  measure 
around. 

§  Radius  is  a  Lat.  word,  from  radio,  to  sJuoot  beams  of  light, 
etc.  The  use  of  this  term  in  geometry,  originates  from  the 
fact,  that  when  a  large  number  of  radii  are  drawn  in  a  circle, 
the  circle  resembles  the  sun  darting  his  rays  in  every  direc- 
tion from  the  center. 


240  RAINEY'S  IMPROVED  ABACUS. 

center  to  the  circumference,  or  half  the  diameter ;  two 
or  more  of  these  lines  are  called  radii. 

The  perimeter*  of  a  circle,  or  other  figure,  is  the  ex- 
tent of  its  circumference  or  bounds. 

The  arecfi  of  a  circle,  or  other  figure,  is  the  surface 
or  space  contained  within  the  limits  of  the  circumfe- 
rence or  perimeter. 

A  circle  is  said  to  be  inscribed  J  in  a  polygon,  when 
the  line  of  the  circumference,  cuts  the  sides  of  the 
polygon. 

A  polygon  is  inscribed  in  a  circle,  when  its  angles 
coincide  with  the  circumference. 

A  polygon  is  circumscribed||  about  a  circle,  when 
its  sides  coincide  with  the  circumference. 

A  semicircle^  is  a  half  circle,  described  by  cutting 
the  circumference  of  a  circle  by  a  right  line  drawn 
through  its  diameter. 

An  ellipse**  is  an  oblong,  circular  figure,  having 
two  axes;  the  minor,  a  transverse,  and  the  major,  a 
longitudinal  line,  each  drawn  through  the  center,  and 
on  either  of  which,  the  figure  may  be  supposed  to 
revolve. 

*  Perimeter  is  from  the  Gr.  mp  or  peri,  around,  about,  par- 
ticularly around  the  space  described  from  a  center,  and  jmtrpsv 
or  metron,  to  measure. 

f  Area  is  a  Latin  word,  which  means  space  within  given 
bounds.  Dr.  Webster  thinks  it  is  from  the  Chaldee  word 
ariga,  a  bed;  or  from  a  Hebrew  word  which  signifies  to 
stretch  or  spread  The  plural  of  area  is  area;  this  is,  however, 
seldom  used;  being  substituted  by  the  anglicised  word,  areas. 

t  Inscribe  is  from  the  Latin  inscribo,  to  write  within. 

II  Circumscribe  is  from  circum,  around,  and  scribo,  to  write 
or  draw. 

IP  Semicircle  is  from  the  Lat.  semi,  half  and  circulus,  a  circle. 

**  Ellipse  is  from  the  Gr.  root  &.KWTOO  or  dleipo,  to  pass  by 
or  reject. 


QUADRATURE    OF    THE    CIRCLE.  241 

The  circumference  of  a  circle  is  divided*  into  360 
equal  parts,  called  degrees  ;  each  of  these  degrees,  into 
60  parts,  called  minutes  ;f  and  each  of  these  minutes 
into  60  parts,  called  seconds.  The  degree,  is  marked, 
thus  (  °  ) ;  the  minute,  thus  ( ' ) ;  the  second,  thus 
(  "  ).  Sometimes  30  degrees  are  said  to  make  1  sign, 
marked  (5);  and  12  signs,  1  circle,  marked  (c). 

QUADRATURE   OF   THE   CIRCLE. 

The  circle,  from  the  varied  nature  of  its  uses  and 
application,  is  one  of  the  most  interesting,  and  yet 
perplexing,  figures  in  geometry.  The  comparison  of 
circles  and  squares;  the  difficulty  of  determining  the 
ratio  of  the  circumference  to  the  diameter ;  the  pro- 
blem of  ascertaining  the  precise  area ;  and  the  diffi- 
culty of  a  continued  application  of  its  principles  to 
the  measurement  of  solid  bodies,  in  the  form  of 
spheres,  cones,  etc.,  have,  in  all  ages,  rendered  its 
study  peculiarly  interesting  to  mathematicians. 

We  shall  consider,  first,  the  relation  of  circumfe- 
rence and  diameter;  next,  the  area;  and  after  this, 
apply  these  principles  to  a  great  variety  of  practical 
measurements. 

The  difficult  and  impossible  problem  of  the  quad- 
rature* of  the  circle,  is  the  determination  of  the  area 
of  a  circle,  whose  diameter  is  equal  to  that  of  a  given 
square,  or  of  a  circle  inscribed  in  a  square.  The  in- 
vestigation of  this  problem  commenced  with  Archime- 
des, a  Grecian.  The  first  step  to  be  taken,  was  evi- 

*  The  division  of  the  circle  into  360  parts,  originated  from 
the  division  of  the  year  by  the  ancients,  into  360  days.  The 
12  signs  represent  the  12  months. 

f  Minute  is  derived  from  the  Lat.  minutum,  a  small  part: 
second,  from  secundus,  the  second,  or  second  order  of  minutes. 

t  Quadrature  is  from  the  Lat.  quadratura,  squaring,  from 
quatuor ',  four ;  reducing  the  circle  to  a  similar  area  of  four 
equal  sides. 


242  RAINEY'S  IMPROVED  ABACUS. 

dently  to  ascertain  the  ratio  of  the  circumference  to 
the  diameter.  And  here,  the  investigation  must  be 
commenced,  by  every  geometer.  It  is  impossible  to 
give  the  process  and  reasoning  used  to  ascertain 
this,  in  a  treatise  on  arithmetic ;  we  will,  however,  in- 
dicate the  process,  and  avail  the  benefits,  without  fur- 
ther investigation.  Neither  the  exact  ratio  of  the 
circumference  to  the  diameter,  nor  the  exact  area  of  a 
circle,  can  ever  be  ascertained ;  and  both  have  long 
since  been  abandoned,  as  impossible,  by  all  good  math- 
ematicians. 

The  method  used,  is  to  inscribe  and  circumscribe 
the  circle  with  regular  polygons ;  then,  to  increase  the 
number  of  the  sides  of  botli  of  these,  to  such  an  ex- 
tent, that  they  seem  to  merge  into  a  common  line ; 
and  although  the  sides  can  never  entirely  coincide,  yet 
they  so  far  coincide,  as  to  give  almost  entire  accuracy, 
to  all  practical  operations.  The  reason  why  they  can- 
not coincide  is,  that  a  curved  and  a  straight  line  can 
never  become  one  line,  however  small  the  degree  of 
curvature.  The  line  of  the  circle  is  always  found 
between  these  two  polygons ;  and  when  the  sides  are 
increased  to  a  very  large  number,  the  human  eye,  as- 
sisted by  the  microscope,  is  unable  to  see  more  than 
one  line  in  the  three.  Archimides  carried  the  number 
of  sides  to  32768,  and  thus  secured  the  ratio  to  seven 
decimal  places.  He  obtained  the  ratio  3^--°-  and  3|3-, 
which,  reduced  to  an  improper  fraction,  gave  2T2,  or 
the  circumference  to  the  circle,  as  22  to  7.  These 
numbers  may  be  used  for  all  ordinary  and  rough  pur- 
poses ;  but  are  far  from  being  accurate,  when  com- 
pared with  the  ratio  afterward  obtained  by  Metius,  a 
German,  who  carried  it  to  17  places  of  decimals,  giv- 
ing the  ratio  of  355  to  113.  Yan  Ceulen,  a  Dutch 
mathematician,  carried  it  yet  much  further,  and  ascer- 
tained that  if  the  circle  was  1,  the  circumference 
would  be  greater  than  3.14159265358579323846264- 


CIRCUMFERENCE    AND    DIAMETER. 

338327950288,  and  less  than  3.14159265358579323- 
846264338327950289,  demonstrating  the  coincidence 
of  the  two  polygons  to  a  fraction  less  than  one  nonil- 
lionth  ;  a  difference  too  small  to  be  adequately  concei- 
ved. The  upper  number  represents  the  inscribed,  and 
the  lower,  the  circumscribed  polygon,  between  which  we 
may  vainly  seek  the  line  of  the  circle.  Later  math- 
ematicians have  carried  the  calculation  as  far  as  140 
places  of  decimals. 

The  relation  of  the  circumference,  may,  from  the 
above,  be  safely  set  down  as  3.141592,  This  gives 
the  exact  ratio  to  5  places  of  decimals,  leaving  a  frac- 
tion as  small  as  TO-O£O'O"O •  ^or  a^  practical  purposes, 
3.1416  may  be  used,  changing  the  5th  decimal,  9,  into 
6,  in  the  4th  order. 

What  is  the  circumference  of  a  circle,  whose  diam- 
eter is  31  feet? 

3.1416x31=97.3836  Ans. 

Here,  we  multiply  by  the  decimals,  and  consequent- 
ly cast  off  four  decimals  in  the  result.  Hence, 

To  find  the  circumference  of  a  circle,  when  the  di- 
ameter is  given^  multiply  the  diameter  by  3.1416,  and 
•cut  off  four  places  of  decimals  in  the  -answer:  or,  for 
entire  accuracy,,  multiply  by  3.1415926,  and  cut  off 
seven  places  for  decimals  at  the  right. 

It  is  desired  to  place  40  sentinels  around  a  camp, 
which  is  1  mile  in  diameter ;  how  far  will  they  be 
apart  ? 

[0-44 


3.1416 


138.2304 


We  reduce*  the  mile  to  yards,  and 
get  the  answer  in  yards ;  they  will  be 
placed  over  138  yards  apart. 

To  find  the  diameter  of  a  circle,  when  the  circum- 
ferejice  is  given,  divide  the  circumference  by  3.1416. 

Or,  multiply  the  circumference  by  7,  and  divide 
%22. 

Qr9  multiply  the  circumference  by  .3183L 


244  RAINEY'S  IMPROVED  ABACUS. 

The  circumference  of  a  circle  is  1  multiplied  by 
3.1416;  hence,  the  diameter  is  1  divided  by  3.1416, 
equal  to  .31831,  which  multiplied  into  the  circumfe- 
rence gives  the  diameter. 

The  circumference  of  a  lot  is  500  yards,  and  it  is 
desired  to  plant  10  trees  on  the  line  of  its  diameter; 
how  far  apart  will  they  be  ? 


The  trees  will  be  placed  nearly  16 
feet  apart. 


.31831 


15.91550 


It  will  be  found  far  more  convenient  to  multiply  by 
this,  than  to  divide  by  the  other  decimal. 


THE    AREA   Of   THE   CIRCLE. 

In  treating  of  polygons,  it  has  been  shown,  that  to> 
ascertain  the  area,  multiply  the  half  radius  by  the  en- 
tire perimeter.  It  has  since  been  shown,  that  the  cir- 
cumference or  perimeter  of  a  circle  includes  a  vast 
number  of  polygons.  Hencey 

To  find  the  area  of  a  circle,  when  the  circumference 
and  diameter  are  given,  multiply  the  circumference 
by  half  the  radius : 

Ory  Place  the  circumference  and  diameter  on  the 
right  of  the  line,  and  4  on  the  left. 

A  circle  is  10  feet  in  diameter,  and  31.416  ft.  m 
circumference ;  what  is  the  area  ? 

One  fourth  of  the  diameter,  equal  to  the  semi- 
radius,  is  2-J  feet,  and  31.416x2^=78.540;  thus, 

#44^0—15708 

5  The  answer  is  78,54  feeL 

T8J4ft  r 


AREA    OF    THE    CIRCLE.  245 

Let  us  suppose  a  square  with  a  circle  of  equal  diam- 
eter inscribed :  the  diameter  of  the  circle  is  equal  to 
the  diameter  of  the  square :  ^  of  the  perimeter  of  a 
square  or  circle  is  the  side  of  such  square  or  circle, 
which  multiplied  into  itself  will  produce  the  area  de- 
noted by  the  perimeter.  The  square  of  the  diameter 
indicates  the  area  of  the  square ;  and  3.1416,  the  area 
of  the  circle :  now,  as  ^  the  perimeter  of  the  square 
is  equal  to  the  side  of  the  square,  so  ^  the  perimeter 
of  the  circle,  .7854,  is  equal  to  the  side  of  the  circle. 
Hence,  if  the  square  of  the  diameter  be  multiplied  by 
•J  of  the  circumference  of  the  circle,  or  .7854,  the  re- 
sult will  be  the  contents  of  the  circle.  For  this  rea- 
son, geometers  take  the  £  of  3. 141592=. 7854,  and 
multiply  it  into  the  square  of  the  diameter,  when  the 
circumference  is  given.  Hence,  the  circle  is  .7854  of 
a  circumscribed  square.  That  is,  if  the  square  is  1, 
the  circle  is  .7854;  or,  if  the  square  contains  10,000 
parts,  the  circle  contains  7,854  parts.  Hence, 

To  find  the,  area  of  a  circle,  when  the  diameter 
only  is  given,  Multiply  the  squared  of  the  diameter 
by  .7854,  and  cut  off  four  places  in  the  answer  for  deci- 
mals. For  greater  accuracy,  multiply  by  .785398, 
and  cut  off  six  places  : 

Or,  Place  the  diameter  and  11  on  the  right ,  and 
14  on  the  left  of  the  line. 

What  is  the  area  of  a  circle  whose  diameter  is  18 
inches  ? 

Here,  18xl8x.7854=254.4696.  The  answer  is 
254^  inches,  nearly. 

How  many  acres  in  a  circular  field  40  rods  in  di- 
ameter ? 

*The  square  of  the  diameter ,  is  the  diameter  multiplied  into 

itself. 


246 


RAINEY'S  IMPROVED  ABACUS. 


.7854 


_ 


The  field  contains  nearly  8  acres.    We 
divide  by  160,  because  this  number   of 
I  square  rods  equals  an  acre. 


What  is  the  area  of  a  circle 
ameter? 


of    an  inch  in  di- 


1 
1 

.7854 


1.19635 


The   answer    is  very   nearly 
square  inch. 


The  cylinder  of  a  steam  engine  is  15  inches  in  di- 
ameter ;  what  is  the  area  in  inches  and  in  feet  ? 


15xl5x.7854=176 
.715  inches. 


Ans. 


98175 


1.7854 


1.22718+ in  ft. 

How  many  acres  of  land  in  a  tract  4000  rods  in  di- 
ameter ? 

Here,  160  rods  opposite,  instead 
of  making  1  acre,  as  would  be  the 
case  in  a  square  tract,  make  in  the 

circular  tract  only  .7854 of  an  acre; 

.  ,.  i 

in  stating,  we  say,  how  many  acres 

will  4000x4000  rods  make,  if  160  rods,  make  .7854 
of  an  acre?  The  answer  is  78,540  acres.  We  may 
ask, 

What  will  the  above  land  cost  at  62i  cents  per 
acre? 

We  may  make  this  among  the  other  statements,  in- 
stead of  separately ;  thus, 


THE    ELLIPSE. 


247 


In  this  instance,  1  acre  is 
placed  opposite  .7854  of  an 
acre,  and  the  price  which  the  1 
acre  equals  last  on  the  right. 


125 


49087.500000 


How  many  inches  in  a  valve  11 J  in  diameter  ? 

895 
4—^95 


Although  very  few  figures 
can  be  canceled  in  this  ques- 
tion, yet  there  is  a  decided  ad- 
vantage in  locating  the  frac- 
tions on  the  vertical  line. 


Ans. 


110.7536+ 


To  ascertain  the  area  of  a  circle,  when  the  circum- 
ference only  is  given,  multiply  the  square  of  the  cir- 
cumference by  the  square  of  7,  and  the  product  by 
.7854;  and  divide  by  the  square  of  22. 

What  is  the  area  of  a  circle  33  ft.  in  circumference. 


This  is  an  easy  and  simple  method 
of  arriving  at  the  result,  by  one 
statement.  The  answer  is  86-|  feet. 
See  Table  of  Circles  and  Areas. 


33 
33 

22 

7 

22 

7 

.7854 

Ans. 

86.59085 

TO   FIND   THE    CIRCUMFERENCE   OF   AN    ELLIPSE. 

Multiply  the  square  root  of  half  of  the  sum  of  the 
two  axes  squared,  by  3.1416;  the  product  will  be  the 
circumference. 

The  axes  are  the  longest  diameters  from  side  to  side, 
and  from  end  to  end. 


TO   FIND    THE   AREA   OF   AN   ELLIPSE. 

Multiply   the  product    of   the   two    diameters    by 

.7854. 


248  RAINEY'S  IMPROVED  ABACUS. 

TO    FIND    THE    CONTENTS    OF   A    SQUARE   INSCRIBED    IN   A 
CIRCLE. 

We  shall  endeavor  to  prove  that  such  a  square  is 
one  half  of  another  square  circumscribed  about  a  cir- 
cle; thus, 

Let  us  draw  a  square  of  a  given  side,  and  inscribe 
in  it  a  circle ;  then,  in  the  middle  of  the  sides  of 
the  square,  where  the  circumference  of  the  circle  cuts 
the  square,  let  us  locate  the  four  angles  of  a  square 
inscribed  in  the  circle.  Thus,  each  side  of  this  in- 
scribed square  will  be  the  hypotenuse  of  a  right-angled 
and  equal- sided  triangle.  Now,  from  the  opposite  an- 
gles of  the  inscribed  square,  draw  two  diagonals.  It 
will  now  be  perceived,  that  the  inscribed  square  con- 
tains four  equal  triangles,  and  the  circumscribed 
square,  eight :  hence,  the  inscribed  square  is  equal  to 
half  of  the  circumscribed  square. 

It  is  manifest,  that  if  we  square  the  semiradius  of 
this  circle,  it  will  equal  two  of  these  triangles ;  hence, 
twice  the  semiradius,  will  equal  four  of  them,  which 
are  equal  to  the  inscribed  square,  or  half  of  the  cir- 
cumscribed square. 

What  is  the  area  of  a  square  inscribed  in  a  circle, 
10  feet  in  diameter? 

10  Here,  we  place  the  square  of  the  diam- 

$  4.0 — 5     eter  on  the  right,  and  2  on  the  left.     This 

"    KA  fl~     is  identical  with  the  other  operation,  in  which 
50  ft.      ,,  .     -,.  j     ,,' 

the  semiradius  would  be  used ;  thus,  semi- 
radius,  5;  this  squared  and  multiplied  by  2,  thus, 
5X£>X2=50  feet,  the  answer,  as  before.  Thus,  by 
the  second  operation,  we  find  it  troublesome  to  find 
the  semiradius,  then  square  it,  and  afterward  multiply 
it ;  while  the  other  process  is  quite  simple  and  easy. 

How  many  cubic  feet  in  a  log  3  feet  in  diameter, 
and  40  ft.  long? 


INSCRIBED    AND    CIRCUMSCRIBED    SQUARES.     249 


Here,  the  perimeter  is  simply  multi- 
plied by  the  length  and  divided  by  2, 
because  the  inscribed  is  one  half  of  the 
circumscribed  square.  We  ask  again, 


— 2 


100 


$',30.00 


180  feet. 

What  will  the  same  come  to  at  $20  per  100  ft.? 

3 
3 

Here,  the  statements  are  combined,  and 
the  answer  is  obtained  in  dollars. 

'~$36 

How  much  will  a  log  cost,  at  33^-  cents  per  foot, 
which  is  18 inches  in  diameter,  and  80ft.  long? 

In  this  instance,  two  twelves  are  0 — JL&  4-$ — 3 
placed  on  the  left,  because  the  two  di- 
mensions on  the  right  are  inches,  and 
must  be  reduced  to  feet.  The  2  is 
placed  on  the  left  as  usual ;  while  33-J-, 
the  price  of  1  foot,  is  placed  on  the 
right;  hence, 

To  find  the  contents  of  a  square  inscribed  in  a  cir- 
cle, place  the  square  of  the  diameter  on  the  right,  and 
2  on  the  left. 

To  ascertain  the  solid  contents,  place  the  length  on 
the  right  likewise. 

To  find  the  price  of  the  whole,  place  the  price  last 
on  the  right)  and  the  quantity  which  it  equals,  on  the 
left. 

TO    FIND     THE     SIDE     OF     A     SQUARE     INSCRIBED     IN     A 
CIRCLE. 

By  reverting  to  the  figure  just  used,  we  find  that 
the  side  of  the  inscribed  square  is  the  hypotenuse  of  a 
right-angled  and  equal-sided  triangle;  .hence,  the 
square  root  of  the  sum  of  the  squares  of  these  two  sides, 


250  RAINEY'S  IMPROVED  ABACUS. 

or,  which  is  the  same  thing,  the  square  root  of  twice 
the  square  of  the  radius,  is  equal  to  the  hypotenuse, 
or  side  of  the  inscribed  square.  This  being  trouble- 
some to  attain,  and  knowing  that  the  smaller  square  is 
half  of  the  larger,  we  extract  the  square  root  of 
3.141592,  which  is  .707106,  and  multiply  it  into  the 
diameter  of  the  circle  for  the  side  of  the  inscribed 
square:  likewise,  we  extract  the  square  root  of  the 
product  of  1  divided  by  3.1416,  into  .707106,  which 
is  .22508 ;  this  multiplied  into  the  circumference  will 
give  the  side. 

How  large  a  square  can  be  hewn  from  a  round 
stick  of  timber,  20  inches  in  diameter? 

Thus,  20 X- 7071=14.142.  The  side  is  14  inches 
and  a  fraction. 

What  is  the  side  of  a  square  that  can  be  sawed 
from  a  log  60  inches  in  circumference  ? 

60  X- 22508— 13.5048  inches,  Ans.  We  may  caU 
the  last  decimal  .2251,  instead  of  .2250,  etc.  Hence, 

To  find  the  side  of  a  square  inscribed  in  a  circle, 
multiply  the  diameter  by  .7071,  or  the  circumference 
by  .2251. 

THE   SIDE   OF   A   SQUARE   GIVEN,    TO    FIND     THE     DIAME- 
TER  OF   THE    CIRCUMSCRIBED    CIRCLE. 

By  reverting  to  the  same  figure,  as  above,  we  find 
that  the  diagonal  of  the  inscribed  square,  is  the  hy- 
potenuse of  a  right-angled  and  equal-sided  triangle, 
whose  sides  are  the  sides  of  the  square.  Hence,  the 
square  root  of  the  sum  of  the  squares  of  these  two  sides 
would  be  the  diagonal,  hypotenuse,  or  diameter.  This 
being  tedious,  we  find  a  decimal,  which  multiplied  into 
the  side,  will  give  the  diagonal  of  the  square,  or  di- 
ameter of  the  circle.  This  decimal  is  1.4142.  Like- 
wise, the  side  multiplied  by  the  decimal  4.443  will 
give  the  circumference  of  the  circumscribed  circle. 


INSCRIBED    AND    CIRCUMSCRIBED    SOTARES.     251 

The  former  of  these  numbers  is  the  square  root  of 
twice  3.141592 ;  and  the  latter,  the  square  root  of  twice 
this  number,  multiplied  by  itself. 

How  large  must  a  tree  be,  in  diameter,  to  square  12 
inches  ? 

12  X  1.4142=16.9704  inches,  Ans. 

How  large  is  the  circumference  of  a  tree,  or  circle, 
around  a  beam  or  square,  whose  sides  are  20  inches  ? 
thus, 

20x4.443— -88.86  inches,  Ans. 

The  sides  of  a  sill  must  be  12-J-  inches ;  how  large 
must  the  tree  be,  in  diameter,  from  which  it  is  sawed? 

By  this  statement,  we  avoid          #25 
the  difficulty  of  using  the  frac- 
tions.    Hence, 


Ans.  17. 5525  inches 

To  find  the  diameter  or  circumference  of  a  circle 
circumscribed  about  a  square  whose  sides  are  given ; 
or,  to  find  the  diameter  of  a  tree,  that  will  square 
a  given  size,  multiply  the  side  of  the  square  by  1.4142 
for  the,  diameter ;  and  by  4.443  for  the  circumference. 
or  girt. 

Jt  is  frequently  necessary  to  find  the  side  of  a  square,  whose 
area  is  equal  to  that  of  a  given  circle;  or  the  diameter  or  cir- 
cumference of  a  circle  whose  area  is  equal  to  that  of  a  given 
square.  As  it  belongs  to  geometry  to  demonstrate  the  vari- 
ous relative  proportions  of  figures,  we  shall  pursue  this  course 
no  further  than  is  demanded  by  a  common-sense  view  of  the 
subject,  and  give  a  few  numbers  without  tracing  their  origin. 

To  find  the  side  of  a  square  whose  area  is  equal  to 
the  area  of  a  given  circle,  multiply  the  diameter  of 
the  circle  by  .8862,  or  the  circumference  by  .2821. 

To  find  the  diameter  or  circumference  of  a  circle, 
whose  area  is  equal  to  the  area  of  a  given  square, 
multiply  the  side  of  the  square  by  1.128  for  the  di- 
ameter, and  by  3.545,  for  the  circumference. 

These  rules  will  be  found  useful  to  mechanics  and 
other  business  men  generally,  and  are  giyen  in  such 
form  as  to  be  easily  understood  and  used. 


252  RATNEY'S  IMPROVED  ABACUS. 

APPLICATION  OF  THE  CIRCLE  TO  CISTERNS.* 

Cisterns  are  large  vaults  made  underground  to  hold 
rainwater.  They  are  of  various  shapes,  square, 
round,  and  conical  or  pyramidal.  A  conical  cistern  is 
round,  but  of  different  diameters  at  bottom  and  top. 
A  pyramidal  is  square,  and  wider  at  bottom  or  top. 

All  that  is  necessary  in  finding  the  contents  of  cis- 
terns, is  to  ascertain  the  number  of  gallons  in  a  solid 
foot,  and  multiply  the  number  of  feet  by  it.  This  is 
done,  by  dividing  the  number  of  inches  in  a  cubic,  or 
cylindric  foot,  by  the  number  of  inches  in  a  wine  or 
beer  gallon. 

A     cubic    foot  contains  7.48    wine  gallons. 

«          "         "         6.127  beer 

"   cylindric   "         "         5.875  wine 

4.812  beer 

The  numbers  5.875,  and  4.812  were  obtained  by 
multiplying  1728,  the  number  of  cubic  inches  in  a  cu- 
bic foot,  by  .7854,  to  reduce  them  to  the  number  of 
inches  in  a  circular  or  cylindric  foot;'  the  product, 
then  divided  by  231  and  282,  respectively,  gave  the 
numbers,  as  above.  Thus,  we  see  the  continued  ap- 
plication of  the  decimal  .7854. 

Were  this  process  not  pursued,  and  the  number  of 
gallons  in  a  foot  obtained,  it  would  be  necessary  to  find 
the  number  of  cubic  inches  in  a  cistern,  and  multiply 
by  .7854,  when  round,  and  divide  by  the  number  of 
inches  in  a  gallon. 

We  may  dispense  with  the  use  of  the  decimal,  par- 
tially, and  multiply  by  two  other  numbers  in  the  form 
of  a  common  fraction.  This  is  done,  by  multiplying 
the  decimal  by  some  number  that  will  terminate  in  ci- 
phers, and  dividing  by  the  same.  For  instance: 

*  The  root  of  cistern  is  the  Lat.  cista,  a  box,  whence  cistcrna, 
a  vault  for  rainwater. 


MEASUREMENT    OF    CISTERNS.  253 

7.48x2=14.96.  Here,  14  is  the  whole  number,  and 
the  .9,  or  T\,  at  the  right  of  it  being  very  nearly  a 
unit,  we  carry  it  to  14  and  call  the  multiplier  15.  We 
divide  this  by  2,  on  the  left  of  the  vertical  line,  as 
well  as  multiply  by  it,  to  make  the  other  number  15. 

A  square  cistern  contains  100  cubic  feet ;  how  many 
wine  gallons  does  it  contain? 

Here  100x7.48=748.00  gallons.  By  the  other 
process,  we  place  the  square  of  the  side  on  the  right, 
multiply  by  15,  and  on  the  left,  divide  by  2 ;  thus, 

It  is  seen  here,  that  the  answers 
vary  only  2  gallons  in  750.  Hence, 
it  is  sufficiently  accurate  for  ordinary 
purposes.  When  strict  accuracy  is  '  Z77[  i7~ 
desired,  the  decimal  may  be  used.  I  Ans'\™"  galls- 
Therefore, 

To  find  the  contents  of  cisterns  which  are  square, 
place  the  square  of  the  side,  and  depth  in  feet)  with 
15.  on  the  right,  and  2  on  the  left,  for  wine  gallons; 
or  multiply  the  number  of  cubic  feet  in  the  cistern  by 
7.48.  For  beer  gallons,  place  49  on  the  right  and  8 
on  the  left,  or  multiply  by  6.127.  For  circular  cis- 
terns, 47  on  the  right  and  8  on  the  left,  or  multiply  by 
5.875,  for  wine  gallons:  and  for  beer  gallons,  24  on 
the  right,  and  5  on  the  left ;  or  multiply  by  4.812. 

For  conical  or  pyramidal  cisterns,  add  the  areas  of 
the  two  ends :  multiply  the  two  areas,  and  extract  the 
square  root:  add  this  to  the  sum  of  the  two  areas 
above :  place  this  sum  and  the  depth  on  the  right,  and 
8  on  the  left :  place  likewise,  on  the  right  and  left,  the 
numbers  representing  gallons,  in  the  measure  desired, 
and,  for  square  or  circular  cisterns,  as  the  caw 
may  be 

It  may  be  remarked,  that  wine  measure,  or  231  cu- 
bic inches  to  the  gallon,  is  generally  used  as  the  stan- 
dard in  the  United  States. 
17 


254 


RA1NEY  S    IMPROVED    ABACUS. 


How  many  wine  gallons  in  a  circular  cistern  12  ft. 
in  diameter  and  20  feet  deep? 


12 

4$  —  3 


4  —  #|47 


The  same  answer  might  be  ob- 
tained by  suspending  the  47  and 
8,  and  placing  5.875  on  the  right. 

Ans]  1692  galls.  ' 

How  many  hogsheads  of  water  m  a  circular  cistern, 
30  feet  in  diameter,  and  21  ft.  deep,  beer  measure  ? 

#0 — 6  In  this  instance,  the  answer 

is  reduced  to  hogsheads,  by 
placing  63,  the  number  of  gal- 
lons in  a  hogshead,  on  the  left. 

How  many  beer  gallons  in  a  circular  cistern  of  con- 
ical shape,  which  is  9  ft.  in  diameter  at  the  top,  7,  at 
the  bottom,  and  9  ft.  deep? 

9x9=81  area  of  upper  end. 

7x7=49     "     "   lower    « 


£ 


Ans. 


24 


1440  hhds. 


i 

5 

193 

0-3 
24 

5 

13896 

Ans. 

27791 

We  first  square  each  of  the  diame- 
ters, and  add  them  ;  next,  multiply  to- 
gether these  squares,  and  extract  the 
square  root ;  we  add  this  root,  63,  to 
the  sum  of  the  two  squares  above,  130, 
making  193 :  this  193,  mean  area,  is 
placed  on  the  line,  with  the  depth  of  the  cistern,  while 
3  is  placed  on  the  left.  This  is  equivalent  to  mul- 
tiplying the  mean  area  by  -J  of  the  depth.  Lastly,  we 
place  24  on  the  right  and  5  on  the  left,  which  numbers 
represent  both  the  number  of  gallons  in  a  solid  foot, 
and  the  deduction  for  the  quadrature  of  the  circle. 
Any  conical  or  pyramidal  cistern  may  be  measured  in 
the  same  way.  It  is  wholly  unnecessary  above  to  get 


DIMENSIONS    AND    CONTENTS    OF    CISTERNS.       255 


first,  the  circular  area  by  multiplying  by  .7854 ;  as 
this  can  be  done  quite  as  easily,  by  the  24  and  5,  or 
by  the  number  which  represents  the  gallons  and  quad- 
rature, 4.812 ;  for  circles  and  squares  are  to  each  other 
as  the  squares  of  their  diameters. 

Required  the  depth  of  a  rectangular  cistern,  to  hold 
5400  gallons,  which  is  8  ft.  wide,  and  10  ft.  long. 

We  here  place  the  contents  of  the  cistern  on  the 
right  of  the  line,  and  the  two  given  dimensions,  with 
the  fraction  expressing  the  number  of  gallons  in  a  foot, 
on  the  left.  The  fraction  in  this  case  is  y ,  the  mea- 
sure being  wine  gallons  :  thus, 

Were  this  beer  gallons,  we 
would  place  4¥9  on  the  left.  If 
the  cistern  were  circular,  we  would 
place  on  the  left  4¥7,  or  2-/,  as  the 
case  might  be,  with  the  square  of 


15 


.Ans. 


££00—135 


9  feet  deep. 


the  diameter,  to  find  the  depth,  or  the  depth,  to  find 
the  square  of  the  diameter.  As  the  answer,  in  the 
latter  case,  would  be  the  square  of  the  diameter,  it 
would  be  necessary  to  extract  the  square  root  for  the 
diameter  of  the  cistern,  Example : 

What  is  the  diameter  of  a  circular  cistern  which 
contains  960  gallons,  beer  measure,  the  depth  being 
8  feet? 

The  result  is  25 :  now,  the 
square  root  of  this,  5,  is  the  di- 
ameter of  the  cistern.  Hence, 


When  two  dimensions  and  the  contents  of  a  cistern 
are  given  to  find  the  other  dimension,  for  rectangular 
cisterns,  place  the  contents  on  the  right>  and  the  two 
given  sides,  with  the  fraction  representing  the  num- 
ber of  gallons  in  a  solid  foot,  on  the  left :  the  answer 
will  be  the  required  side. 

For  circular  cisterns,  place  the  contents  on  the 
right,  and  the  square  of  the  diameter  and  fraction,  on 


256 


RA1NEY  S    IMPROVED    ABACUS. 


the  left,  for  the  depth ;  or  the  depth  and  fraction,  for 
the  square  of  the  diameter  ;  in  the  latter  case,  extract 
the  square  root,  and  the  answer  will  be  the  required 
diameter. 

TABLE, 

Giving  the  capacity  of  Square  and  Circular  Cisterns, 
Wells,  etc.,  1  foot  deep,  in  wine  and  beer  gallons. 


Diam., 
or  side 
in  feet. 

Circular   Cisterns. 

Square    Cisterns. 

The  table  of  cis- 
terns  of    different 
figures,  and  of  dif- 

Wi. gall. 

B'rgall. 

Wi.  galls. 

B'r  galls. 

3 

52.875 

43.2 

67.5 

55.125 

ferent     kinds     of 

3V 

71.96 

58.8 

91.875 

75.031 

measure,  is    given 

4 

994. 

76.8 

120. 

98. 

for  the  purpose  of 

118.969 

146.875 

97.2 
120. 

151.875 

187.5 

124.031 
153.125 

convenience.    The 
calculations  are 

5V 

177.719 

145.2 

226.875 

185.906 

made  for  cisterns  1 

6  2 

211.5 

172.8 

270. 

220.5 

foot    deep;    there- 

gi/ 

248.219 

202.8 

316.875 

258.781 

fore,  nothing  more 

7 

287.875 

235.2 

367.5 

300.125 

is  necessary  in  as- 

7v 

330,469 

270. 

421.875 

344.531 

certaining  the  con- 

§ 

376. 

307.2 

480. 

392. 

tents   of    a    given 

sy 

424.437 

346.0 

541.875 

442.531 

cistern  than  to  find 

9 

475.875 

388.8 

607.5 

496.125 

the  contents  of   a 

9V 

530.219 

433.2 

676.875 

552.781 

given   diameter  in 

10 

587.5 

480. 

750. 

612.5 

the  table,  and  mul- 

!!* 

647.719 

710.875 

529.2 

580.8 

828.875 
907.5 

675.281 
741.125 

tiply  it  by  the  re- 
quired depth.    For 

iji/ 

776.969 

634.8 

991.875 

810.031 

example:  it  is  de- 

12 2 
13 

846. 
992.875 

691.2 
811.2 

1080. 
1267.5 

882. 
1035.125 

sired  to  know  how 
much  a  circular  cis- 

14 
15 

1151.5 
1321.875 

940.8 
1080. 

1475. 

1687.5 

1200.5 
1378.125 

tern  20  ft.  in  diam. 
and  30  ft.  deep,  will 

16 

1504. 

1228.8 

1920. 

1568. 

contain,    in    wine 

17 

18 

1697.875 
1903.5 

1387.2,2167.5 
1555.22430. 

1770.125 
1984.5 

gallons.     We  refer 
to  the  table  and  find 

19 

2120.875 

1732.8 

2707.5 

2211.125 

that  such  a  cistern 

20 

2350. 

1920. 

3000. 

2450. 

1     foot    deep    will 

21 

2590.875 

2116.8 

3307.5 

2701.125 

contain  2350  galls., 

22 

2843.5 

2323.2 

3630. 

2964.5 

and  infer  that    30 

23 

3107.875 

2539.2 

3967.5 

3240.125 

times  this  or  70500, 

24 

3384. 

276£8 

4320. 

3528. 

will    be    the   con- 

25 

3671.87513000. 

4687.5 

3828.125 

tents    of    the    re- 

DEFINITIONS    IN    SPHERICAL    GEOMETRY.        257 
CYLINDERS,    CONES,    SPHERES,    ETC. 

A  cylinder*  is  a  round  solid  body,  of  uniform  diam- 
eter, whose  two  ends  or  bases  are  at  right  angles  with 
the  sides. 

A  cone'f  is  a  round  solid  body,  tapering  in  a  direct 
line  from  the  periphery  of  the  base  to  a  point  at  the 
top,  called  the  vertex. 

A  pyramid^  is  a  square  body,  tapering  in  a  direct 
line  from  the  periphery  of  the  base  to  the  vertex. 

A  frustu7Ji\\  of  a  cone,  or  pyramid,  is  the  part  re- 
maining after  a  portion  of  the  top  is  cut  off  by  a  plane 
parallel  to  the  base. 

The  convex^  surface  of  a  cylinder,  cone,  pyramid, 
or  frustum,  is  the  curved  surface,  exclusive  of  the 
ends,  or  bases.  The  entire  surface,  includes  the  area 
of  the  ends,  or  bases. 

A  sphere^  is  a  solid,  round  body,  with  a  curved 
surface,  all  parts  of  which  are  equally  distant  from  the 
center  within ;  and  is  generated  by  the  revolution  of  a 
semicircle  about  its  own  side. 

A  spheroid**  is  an  oblong  sphere,  whose  diameter 
across  is  less  that  the  diameter  of  the  opposite  ends, 
and  is  formed  by  the  revolution  of  an  ellipse  about 
either  of  its  axes.  The  extremes  of  the  longer  diam- 
eter are  called  the  major  axis,  and  of  the  shorter,  the 

*  Cylinder  is  from  the  Gr.  root  KVXI&  or  kulio,  to  roll. 

t  Cone  is  from  the  Welsh  con,  that  which  shoots  a  point. 

J  Pyramid  originates  from  a  Gr.  word  whose  origin  is  ?rvp 
or  pur,  a  flame,  from  its  resemblance  in  shape  to  a  blaz- 
ing fire. 

||  Frustum  is  a  Lat.  word,  which  means  a  broken  piece. 

§  Convex  is  from  the  Lat.  convexus,  to  bend  down  on  every 
side,  as  the  heavens;  the  opposite  of  concave,  from  cavus,  a 
hollow. 

If  Sphere  is  from  the  Lat.  sphcera,  a  round  ball. . 

**  Spheroid  is  from  the  Gr.  o-v&ip*  or  sphaira,  a  round  ball, 
and  ttcfo?  or  eidos,form;  globe-like. 


258  RAINEY'S  IMPROVED  ABACUS. 

minor  axis,  from  the  Latin,  major  and  minor,  greater 
and  less. 

The  axes*  of  a  body  are  points  on  which  it  is  sup- 
posed to  revolve. 

The  diameter  of  a  sphere  is  a  line  drawn  through  the 
extreme  opposite  surfaces.  The  radius,  a  line  drawn 
from  the  center  to  any  part  of  the  surface. 

A  section,  as  of  a  cone,  or  other  body,  is  a  portion 
cut  off;  from  the  Lat.  root  seco9  to  cut  ojf.f 

THE    CYLINDER. 

To  find  the  convex  surface  of  a  cylinder,  multiply 
the  circumference  by  the  length:  for  entire  contents, 
add  to  this  twice  the  area  of  the  base,  or  the  area  of 
the  two  ends. 

It  may  be  observed  that  the  convex  surface  of  a  cylinder  is 
a  rectangular  figure,  supposing  it  to  be  rolled  out  in 
the  form  of  a  plane.  Hence,  the  length  multiplied  by  the 
width,  will  give  the  superficial  contents,  as  in  other  cases  of 
rectangles. 

What  is  the  convex  surface  of  a  cylinder,  18|-  in. 
in  circumference,  and  8  ft.  long? 

6— 4 


75 


O  1      £        j_ 

24  feet. 


Having  the  length  in  feet,  we  di- 
vide by  12,  to  reduce  the  width  to 
the  same  dimension,  that  the  an- 
swer may  be  in  feet. 

What  is  the  entire  surface  of  a  cylinder  30  inches 
in  circumference,  and  40  inches  long  ? 

*  Axes  is  the  plural  of  axis,  which  is  from  ct%u>v  or  axon,  a 
table  at  Athens  on  which  the  laws  were  written,  and  which 
revolved  on  centers  or  axes.  This  word  is  from  <iyu>  or  ago, 
to  guide  or  direct  movements-,  hence,  the  direction  of  the  mo- 
tion of  a  body. 

fThis  is  the  origin  of  that  beautiful,  yet  very  difficult  de- 
partment of  geometry,  called  Conic  Sections,  whose  demon- 
stration requires  the  closest  and  most  perspicuous  reasoning. 


MEASUREMENT    OF    THE  CYLINDER. 


259 


2 

5 

2 

5 

22 

7 

22 

7 

.7854 

2 

.9939 

area  of  both  ends. 

8.3333 

convex  area. 

12  £0 


feet. 


Ans.  9.3272  entire  surface. 

In  the  work  above,  the  difficulty  is  presented  of 
finding  the  diameter  of  a  circle,  then  squaring  it,  and 
multiplying  by  .7854,  for  the  area,  and  2,  for  the  two 
ends,  which  would  make  two  statements  necessary. 
Hence,  to  avoid  this,  we  multiply  together  the  square 
of  the  circumference,  the  square  of  7,  the  decimal 
.7854,  and  2,  representing  the  two  ends,  and  divide  by 
the  square  of  22,  and  the  denominators  found  in  the 
circumference. 

To  find  the  solid  contents  of  a  cylinder,  place  the 
square  of  the  diameter,  the  length,  and  .7854  on  the 
right,  and  such  denominate  numbers  as  may  be  neces- 
sary, on  the  left:  or,  instead  of  .7854, place  11  on  the 
right,  and  14  on  the  left. 

What  is  the  solid  contents  of  a  cylinder  30  inches 
in  diameter,  and  16  feet  long? 

Here  the  two  twelves  are 
placed  on  the  left  to  reduce  the 
square  of  the  diameter  to  feet, 
that  all  of  the  dimensions  may 
be  in  feet.  Four  places  of  de- 
cimals are  cut  off  in  the  answer. 


Ans. 


.7854 


78.5400  ft. 


How  many  solid  feet  in  a  cylinder  10^  feet  in  di- 
ameter and  12  ft.  long? 


260 


RAINEY'S  IMPROVED  ABACUS. 


21 


ll 


22079 


The  11  and  14  are  used  in  this 
calculation.  These  are  not  suffi- 
ciently accurate  in  cases  where 
great  precision  is  desired. 


CONTENTS   OF   BOILERS. 

To  find  the  contents  of  boilers,  place  the  square  of 
the  diameter  and  length  in  feet,  on  the  right  with  47, 
and  8  on  the  left :  the  answer  will  be  the  contents  in 
wine  gallons.  Ascertain  the  contents  of  the  fiues  in 
the  same  ivay,  and  subtract  these  from  the  contents 
above. 

For  beer  gallons,  instead  of  47  and  8,  use  24 
and  5. 

Or,  Multiply  the  length  in  feet,  by  the  contents  of 
the  given  diameter,  as  found  in  the  subjoined  table, 
and  strike  off  as  many  figures  in  the  answer  for  deci- 
mals as  there  are  decimal  places  in  the  multiplier. 

How  many  wine  gallons  in  a  boiler  39  inches  in  di- 
ameter and  40  feet  long,  having  two  flues,  each  9 
inches  in  diameter? 
43 


47 


Boiler. 


Flues.  264.375 


£—13 


47 


2482.1875 
264.375 


Ans.  2217.8125  gallons. 

In  this  example  we  make  a  separate  calculation  for  the 
flues,  which  are  %  of  a  foot  in  diameter;  and  as  there  are 
two  of  them,  we  multiply  by  2,  instead  of  working  the  ques- 
tion twice.  The  answer  is  2218  gallons,  nearly,  wine  mea- 
sure. The  same  process  may  be  pursued  for  beer  gallons. 

The  subjoined  table  will  be  found  very  convenient  for  prac- 
tical men* 


AIR    PRESSURE,    CONES,  PYRAMIDS,    ETC.       261 

TABLE 

Of  contents  of  Boilers  1  foot  in  length,  in  wine  gallons,  from  3 
to  38  inches  in  diameter. 


Dla. 

Cent's. 

Dia. 

Cent's. 

Di. 

Cent's. 

Di. 

Cent's. 

Di. 

Cent's. 

3 

.367 

7^ 

2-293 

12 

5.875 

21 

17.992 

30 

36.719 

3*X 

.499 

8 

2.611 

13 

6.895 

22 

19.739 

31 

39.207 

4 

.652 

W 

2.947 

14 

7.997 

23 

21.582 

32 

41.777 

4X 

.826 

9 

3.304 

15 

9.179 

24 

23.5 

33 

44.421 

5 

1.02 

9# 

3.682 

16 

10.444 

25 

25.499 

34 

47.163 

5^ 

1.234 

10  ' 

4,079 

17 

11.788 

26 

27.666 

35 

49.978 

6 

1.469 

IQiX 

4.498 

18 

13.219 

27 

29.308 

36 

52.875 

6K 

1.724 

11 

4.937 

19 

14.719 

28 

31.985 

37 

55.853 

7 

2. 

UK 

5.396 

20 

16.319 

29 

34.311 

38 

59. 

AIR   PRESSURE. 

To  find  the  air  pressure  on  piston  heads  and  other  circular 
areas,  place  the  square  of  the  diameter  and  165  on  the  right,  and 
14  on  the  left. 

This  rule  is  based  on  the  supposition  that  the  pressure  of 
air,  per  superficial  inch,  is  15  lbs.,and  that  LI  is  the  ratio  of 
the  circle  to  the  square:  hence,  instead  of  using  the  15  and  11, 
separately,  we  make  them  one  number,  such  as  may  be  easi- 
ly remembered,  by  multiplying  them  together.  The  pressure 
may  be  quite  accurately  obtained  by  multiplying  by  59  and 
dividing  by  5;  though,  for  entire  accuracy,  it  is  best  to  use  1.5 
and  the  decimal  .7854,  which,  multiplied  into  the  square  of  the 
diameter,  will  give  the  true  result 

What  is  the  air  pressure  on  a  piston  head  10-j- 
inches  in  diameter  ? 


The  answer  is  nearly  1300 
pounds. 


Ans. 


21 


i— 3 


165 


10395 


12S9|  Ibs. 

What  is  the  air  pressure  on  a  piston  head  35  inches 
in  diameter,  in  tons  ? 


262 


RAINEY'S    IMPROVED    ABACUS. 


2—2— 400/tf— 7 
2-20  tt—l 


32 


AnS. 


231 


The  answer  is  7^  tons, 
equal  to  7  tons,  and  437-|  Ibs., 
avoirdupois. 


CONES    AND    PYRAMIDS. 


To  find  the  convex  surface  of  a  cone  or  pyramid,  multiply  the 
circumference  or  perimeter  of  the  base  by  the  slant  hight,  and  di- 
vide by  2. 

The  reason  for  dividing  by  2,  is  that  the  convex  surface  of 
a  cone  or  pyramid  is  just  %  the  convex  surface  of  a  cylinder 
of  similar  base  and  altitude. 

The  circumference  of  a  cone  is  90  inches,  and  the 
slant  hight  10  feet ;  how  many  feet  in  its  surface  ? 

J00—3 

Twelve  is  used  on  the  left,  to  re- 
duce the  circumference  to  feet. 


10—5 


150 


Ans.    37|  ft. 

To  find  the  solidity  of  a  cone  or  pyramid,  multiply  the  square 
of  the  diameter,  the  vertical  hight,  and  .7854  together,  and  divide, 
by  3,  and  such  other  numbers  as  may  be  necessary  to  reduce  the 
dimensions  to  the  came  denomination.  Cut  off  four  figures  for 
decimals  in  the  answer. 

The  reason  for  dividing  the  product  of  the  dimensions  by 
3,  is,  that  a  solid  cone  is  one-third  of  a  cylinder  of  similar 
base  and  altitude.  The  same  thing  is  true  of  a  pyramid. 

How  many  solid  feet  in  a  cone  10  feet  in  diameter, 
and  30  feet  high? 

10  The  answer  is   785£   solid  feet, 

X 10  nearly.  We  might  use  the  11  and  14, 

instead  of  .7854,  which  would  give 
.7854  the  same  answer  within  a  very  small 

fraction. 


Ans. 


785.4000 


To  find  the  convex  surface  of  the  frustum  of  a  cone  or  pyramid, 


SOLIDITY  OF  THE  FRUSTUM  OF  A  CONE.   263 

place  the  sum  of  the  circumferences,  or  perimeters,  of  the  two 
ends,  with  the  slant  hight  on  the  right,  and  2  on  the  left;  for  the 
entire  surface,  add  to  the  result,  the  sum  of  the  areas  of  the  two 
ends. 

A  cone,  superficially  measured,  may  be  easily  proven  to 
be  a  triangle;  hence,  the  necessity  of  dividing  the  product  of 
its  two  sides,  as  a  rectangle,  by  2,  to  get  the  triangle.  Like- 
wise, the  superfice  of  a  frustum  may  be  divided  into  two  tri- 
angles: hence,  the  necessity  of  the  sum  of  the  circumfe- 
rences. 

To  find  the  solidity  of  the  frustum  of  a  cone  or  py- 
ramid. To  the  sum  of  the  areas  of  the  two  ends,  add 
the  square  root  of  the  product  of  these  areas;  place 
this  sum,  with  the  vertical  hight,  on  the  right  of  the 
line,  and  3  on  the  left. 

Thus  we  obtain  the  mean  and  two  end  areas,  which  are  the 
bases  of  thrfce  right  cones:  these  three  cones  are  equal  to  the 
solid  frustum. 

How  many  solid  feet  in  the  frustum  of  a  cone  4|  ft. 
high,  and  6  ft.  in  diameter  at  the  lower,  and  5  ft.  at 
the  upper  end  ? 

6x6=36 X- 7854=28.27  area  of  lower  base. 
5 X 5=25 X- 7854=19.63     «     "  upper    " 

47.9     sum  of  bases. 
28.27x19-63=^/554.94=23.55. 

Now,  47.9+23.55=71.45,  and  this  multiplied  by 
the  hight,  and  divided  by  3,  gives  the  answer ;  thus, 

In  the  work  above,  we  ascer-  #  fiji.jif; — 1429 

tained,  first,  the  two  areas,  mul- 
tiplied them,  and  extracted  the 
square  root  of  their  product, 
which  root  was  23.55.  This,  added  to  the  sum  of  the 
two  areas,  47.9,  gave  71.45,  to  be  multiplied  by  -J-  of 
the  hight.  It  is  almost  impossible  to  explain  such 
operations  as  this  satisfactorily  by  arithmetic ;  yet  their 
utility  in  practical  affairs  necessitates  their  intro- 
duction. 


Ans. 


114.32 


264  RAINEY'S  IMPROVED  ABACUS. 

THE  SPHERE  OR  GLOBE. 

To  find  the  surface,  of  a  sphere  or  globe,  Multiply 

the,  circumference  by  the  diameter : 

Or,  Multiply  the  square  of  the  diameter  by  3.1416 : 
Or,  Multiply  the  square  of   the  circumference  by 

.3183. 

The  decimal  .3183  is  obtained  by  dividing  1  by  3.14159. 

What  is  the  surface  of  a  sphere,  whose  diameter  is 
3  feet?  thus, 

3x3=9,  the  square  of  the  diameter;  and  this  9x 
3.1416=28.2744,  or  28  superficial  feet,  the  answer. 

To  find  the  solidity  of  a  sphere, 

Multiply  the  cube*  of  the  diameter  by  the  decimal 
5236  : 

Or,  Multiply  the  square  of  the  diameter,  .7854,  £ 
of  the  diameter,  and  4  together: 

Or,  Multiply  the  square  of  the  diameter  by  3.1416, 
and  the  product  by  ±  of  the  diameter. 

The  decimal  .5236  here  used,  is  one-sixth  of  3.1416.  In- 
deed, all  of  the  decimals  used  as  multipliers  in  anything  per- 
taining to  circular  figures,  can  be  traced  back  to  the  number 
representing  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter. 

A  globe  is  10  inches  in  diameter;  how  many  solid 
inches  does  it  contain? 

10 


10  X 10  X  10=1000  X.5236r= 
523.6  cubic  inches,  Ans.;  or, 


10xlOX.7854xlfX4=523.6;  or,         35 


Ans. 


10 
3.1416 


523.6000 


Any  of  the  processes  above  may  be  used  with  en- 
tire safety. 

*  The  cube  of  a  number  is  that  number  multiplied  into  it- 
self twice;  the  cube  of  4  is  64. 


THE    SPHEROID,    WEIGHT    OF    GLOBES,    ETC.      265 


A  globe  is  30  inches  in  diameter ;  how  many  solid 
feet  does  it  contain  ? 

It  is  frequently  quite  convenient 
to  state  on  the  line,  where  reduc- 
tion is  to  be  performed,  or  any 
fractional  numbers  used.  The 
question  may  be  stated  thus, 


4—  /i£ 
4—^ 
12 

tfO 

£0 
3.1416 

5 

Ans. 

8.18125 

e  are 

2 

5 

imen- 

2 

5 

must 

12 

3.1416 

5 

The  two  dimensions  in  the  square  are 
reduced  to  feet,  while  the  third  dim  en  - 
sion,  in  the  cube,  is  still  inches,  and  must 
be  reduced  by  12. 

It  may  be  remarked,  that  when  a  cylinder  is  circumscribed 
about  a  sphere,  whose  length  is  equal  to  its  own  diameter,  or 
that  of  the  sphere,  the  relation  of  the  surface  of  the  sphere 
to  the  entire  surface  of  the  cylinder  is  as  2  to  3;  and  that  the 
relation  of  their  solidities  is  the  same.  Hence,  an  easy  method 
of  finding  the  solidity  of  a  sphere,  is  to  take  two-thirds  the 
solidity  of  a  cylinder,  of  diameter  and  length  equal  to  tho 
diameter  of  the  sphere. 

THE   SPHEROID. 

To  find  the  solidity  of  a  spheroid, 

Multiply  the  square  of  the  shorter  or  minor  axis  by 
the  longer  or  major  axis,  and  the  product  by  .5236. 

A  spheroid  is  20  by  30  inches;  what  is  its  solidity? 
20X20X30X. 5236— 6283.2  cubic  inches.  Ans. 

Lines  are  to  each  other  as  their  linear  extent: 

Areas  are  to  each  other  as  their  squares :  and 

Solidities  are  to  each  othtr  as  their  cubes.  There- 
fore, 

To  ascertain  the  weight  of  a  globe,  when  the  weight 
of  a  globe  of  similar  material  is  given,  Place  the 
cube  of  the  diameter  of  the  globe  whose  weight  is  re- 
quired, on  the  right,  and  the  cube  of  the  globe  whose 
diameter  is  given,  on  the  left,  and  the  weight  of  the 
given  globe,  on  the  right. 


266  RAINEY'S  IMPROVED  ABACUS. 

The  solidity  of  a  globe,  whose  diameter  is  one  inch,  is 
.52359,  which,  for  practical  purposes,  is  called  .5236^. 

A  globe  of  wrought  iron,  1  foot  in  diameter,  weighs  254.8 
Ibs.,  and  of  cast  iron,  242.  The  weight  of  bar  iron  being  1, 
the  weight  of  cast  iron  is  .95,  of  steel,  1.02,  copper,  1J.6, 
brass,  1.09,  and  lead,  1.48. 

A  cubic  foot  of  rolled  iron  weighs  486.65  Ibs.,  avoirdupois; 
a  cylindric  foot,  382.2  Ibs.  Hence,  a  cubic  inch  weighs 
.28166  of  alb.;  a  cylindric  inch,  .22116;  now,  taking  two 
thirds  of  the  latter,  shows  that  a  spherical  inch  of  rolled  iron 
weighs  .14744,  and  of  cast  iron,  .14006  of  a  Ib.  A  cubic  inch 
of  cast  iron  weighs  .26757  of  a  Ib.;  hence,  3.84  cubic  inches 
of  cast  iron  weigh  1.02  Ib.;  3.84  cubic  inches  are  generally 
allowed  for  1  Ib. 

If  a  cast  iron  globe,  12  inches  in  diameter,  weighs 
242  Ibs.,  how  much  will  a  globe  of  the  same  metal 
weigh,  which  is  15  inches  in  diameter? 

Here,  the  cube  of  the  di- 
ameter is  placed  on  the  line  in 
inches,  in  each  case ;  and  as  the 
cube  of  12  equals  242,  the 
cube  of  15  must  equal  473, 
etc. 


Ans. 


GAUGING.* 

Gauging  is  the  measurement  of  casks  or  barrels ; 
and  is  subject  to  no  specific  rules,  from  the  fact  that  a 
cask  is  not  identical  with  any  regular  geometrical 
figure;  hence,  no  certain  directions  can  be  given  to 
measure  all  of  the  various  shapes  which  casks  assume. 
They  are  generally  considered  the  two  equal  frustums 
of  a  cone,  with  greater  or  less  lateral  curvature. 
It  is  not  necessary  to  enter  into  an  investigation  of 
the  principles  on  which  we  found  the  following 

*  Gauge  is  from  the  French,  jauge,  a  measuring  rod. 


PHILOSOPHICAL    CALCULATIONS.  267 


DIRECTIONS   FOR   GAUGING   CASKS. 

Place  the  sum  of  twice  the  square  of  the  bulge  di- 
ameter and  once  the  square  of  the  head  diameter,  with 
the  length,  on  the  right ;  and,  on  the  left,  882  for 
wine  gallons,  and  1077  for  beer  gallons:  Or,  Multi- 
ply the  square  of  the  mean  diameter  by  the  length, 
and  the  product  by  .0034  for  wine,  and  .0028  for 
beer  gallons. 

To  ascertain  the  mean  diameter  between  the  bung  and  the 
head,  where  the  stave  is  greatly  curved,  add  to  the  head  diam- 
eter .7  of  the  difference  between  the  head  and  bung  diame- 
ters; when  moderately  curved,  .55;  and  when  very  slightly 
curved,  .5. 

How  many  wine  gallons  in  a  cask  49  inches  long, 
30  inches  bung,  and  21  inches  head  diameter?  thus, 

30x30x2=1800 
21x21       =  441 

2241 


6 


747 


Ans.  124^  galls. 

The  measurement  of  casks  by  calculation  is  of  but 
little  utility,  as  it  is  now  mostly  done  by  a  rod,  with 
computations  already  made,  in  tabular  form. 


MECHANICAL  POWERS. 

The  remarks  on  the  mechanical  powers  will  be  very 
limited,  as  this  subject  belongs  legitimately  to  natural 
philosophy.  Yet  we  may  give  such  general  directions 
as  will  enable  the  student  in  philosophy  to  make  his 
calculations  with  greater  facility,  than  by  the  old 
method. 

The  mechanical  powers  are  six :  the  lever,  the  in- 
clined plane,   the   wheel   and  axle,   the    pulley,   the 
screw,  and  the  wedge. 
18 


268  RAINEY'S  IMPROVED  ABACUS. 

Several  of  these  are,  however,  the  same  powers, 
which  receive  their  name  from  the  nature  of  their  ap- 
plication; as  there  are,  strictly  speaking,  but  two 
powers,  the  lever  and  inclined  plane.  The  wheel  and 
axle,  and  pulley  are  revolving  levers ;  the  screw,  a 
revolving  inclined  plane  ;  and  the  wedge,  a  compound 
inclined  plane. 

The  fulcrum*  of  a  lever  is  the  point  or  pivot  on 
which  the  lever  rests:  the  arm  is  the  distance  be- 
tween this  rest  and  the  power  or  weight. 

When  the  two  arms  of  a  Lever  and  the  power  are  given,  to  find 
the  weight  that  will  equiponderate,  proceed  as  in  Inverse  Propor- 
tion: Or,  Place  the  length  of  the  arm  for  which  the  weight  is  re- 
quired, on  the  left,  and  the  other  arm  and  the  given  weight  on  the 
right:  the  answer  will  be  the  required  weight. 

A  lever  20  feet  long  rests  on  a  fulcrum  5  feet  from 
one  end ;  on  the  short  end  is  a  weight  of  3000  Ibs.r 
what  weight  attached  to  the  other  end  will  equipon- 
derate ? 


teooo 


1000  Ibs. 


This  may  be  proven  by  finding 
the  length  of  one  of  the  arms. 


If  the  arm  of  a  lever  15  feet  long,  with  1000  Ibs. 
attached,  equiponderate  3000  Ibs.,  how  long  is  the  arm 
to  which  the  latter  weight  is  attached  ? 

1000  Here,  3000  feet  is  the  demand,  1000 

]Lp — 5  the  same  name,  while  the  term  of  an- 

~T~~  r  f  swer  is  15  feet.  The  1000  Ibs.  and  the 
1  15  cooperate  to  produce  the  common 

effect,    equiponderance.      Hence,  they  are  causes  in 

proportion. 

When  a  weight  is  sustained  between  two  props  or  fulcra,  pro- 
ceed by  Inverse  Proportion;  making  the  entire  length  of  the  lever 
the  demand,  the  sJiort  arm  the  same  name,  when  the  weight  on  the 

*  Fulcrum  is  a  Latin  word  which  means  a  prop,  or  brace. 


MECHANICAL    POWERS. 


269 


prop  of  the  long  arm  is  required,  and  vice  versa,  and  the  whole 
weight  the  term  of  answer. 

Two  men,  A  and  B,  carry  a  burden  on  a  lever  30 
feet  long,  placed  10  feet  from  A;  what  is  the  weight 
sustained  by  B  ? 


30 


20 
400 


30 


10 

400 


Ans.  266f  A's.  Ans.  1.33£  Ibs.  B's. 

The  weight  to  the  answer  is  inversely  as  the  arm  to 
the  whole  lever.  These  two  results  added,  make  400 
Ibs.,  the  whole  weight. 

The  diameter  of  the  wheel,  the  diameter  of  the  axle,  and  the 
power  given,  tofnd  the  weight:  Proceed  as  in  Inverse  Propor 
tion,  etc. 

The  diameter  of  a  wheel  is  21  ft.,  and  that  of  the 
axle  8  inches ;  what  weight  attached  to  the  axle  will 
balance  140  Ibs.  attached  to  the  periphery  of  the 
wheel  ? 

We  here  make  the  8  inches  f  of  a 
foot,  and  placing  it  on  the  left,  di-       100 
vide  likewise  by  100  which  reduces 
the  answer  to  cwts.:  hence,  50|  cwt. 
The  radius   of  a  wheel  is  the  long 
arm;    the  radius   of    the   axle,  the      Ans. 
short  arm.     The  weight  is  the  power. 


10 


21 

440— 7 

504 

50| 


This  question 

may  be  proven  by  finding  the  power  attached  to  the 
wheel,  or  by  finding  the  radius  of  either  the  wheel 
or  axle.  The  ingenious  pupil  may  experiment  at 
pleasure. 

The  length,  the  Jdglit,  and  the  power  of  an  inclined  plane  given, 
tofnd  the  weight:  Make  the  length  the  demand  in  direct  propor- 
tion; the  hight,  the  same  name;  and  the  power,  the  term  of  an- 
swer: the  answer  will  be  the  weight. 

To  find  the  power,  make  the  hight  the  demand,  etc. 

An  inclined  plane  is  72  ft.  long,  and  8  ft.  high ; 
what  weight  will  781  Ibs.  power  sustain? 


270 


RAUTEY'S  IMPROVES  ABACUS. 


7gj  The  demand,  72,  is  placed  on  the 

right.     Again: 

Ans.  7029  Ibs, 

What  power  will  sustain  7029  Ibs.  weight  on  an  in- 
clined plane  72ft.  long,  and  8  ft.  high? 


We  now  get  the  power  assumed 
in-  the  first  question,  for  the  answer. 

The  side  of  a  wedge,  the  thickness  of  the  head,  and  the  power 
given,  to  ascertain  the  force:  Make  the  length  the  demand;  the 
thickness,  the  same  name  ;  and  the  power,  the  term  of  answer. 

The  dimensions  and  resistance  given, to  fnd  the  power:  Make 
the  thickness  the  demand;  the  length  the  same  name;  and  the  re- 
sistance the  term  of  answer. 

The  length  of  a  wedge  is  40  inches,  the  head  8 
inches,  and  the  power  300  Ibs.  what  is  the  force? 

This  may  be  proven  by  finding  the 
power,  the  force  being  the  resistance ; 
thus, 


Ans. 


40 

300 


Ans. 


1500  Ibs. 

:— 2 

1500 


300  Ibs. 


These  statements  are  made  by  di- 
rect proportion. 


The  distance  between  the  threads  of  a  screw,  the  length  of  lever, 
and  power  given,  to  ascertain  the  weight:  Make  the  circumference 
whose  radius  is  the  lever,  the  demand;  the  distance  between  the 
threads,  the  same  name;  and  the  power  the  term  of  answer. 

The  weight  given,  to  ascertain  the  power:  Make  the  distance 
between  the  threads,  the  demand;  the  circumference,  as  above,  the 
same  name;  and  the  power,  the  term  of  answer. 

The  distance,  parallel  to  the  center,  of  a  screw  be- 
tween its  threads,  is  2-J-  inches ;  the  length  of  the 
lever,  56-J-  inches,  and  the  power,  9  tons ;  what  pres- 
sure will  it  give  ? 


SQUARE    ROOT. 


271 


Here,  355  is  the  circumference  of 
twice  the  radius,  56  £. 


5355 

2 
9 


Ans. 


1278  tons. 


Let  us  now  find  the  power,  the  pressure,  1278  tons, 
being  ascertained ;  thus, 

We  merely  reverse  the  supposition 
and  demand,  after  finding  weight,  force, 
pressure,  etc.,  to  find  the  power. 

We  trust  enough  has  been  said  to  render  the  appli- 
cation of  arithmetic  to  philosophy,  plain  and  simple,  so 
far  as  the  mechanical  powers  are  concerned. 


355 

2 

5 

1278 

Ans. 

9  tons. 

SQUARE  EOOT. 

The  extraction  of  the  square  root,  and  all  the  other 
roots,  depends  on  principles  which  it  is  extremely  dif- 
ficult to  explain  satisfactorily,  in  arithmetic.  The 
roots  belong  properly  to  Algebra ;  and  the  only  ex- 
cuse for  the  notice  of  square  root  here,  is,  that  we 
very  frequently  require  it  in  practical  affairs.  Cube 
root,  to  the  contrary,  is  very  seldom  needed,  except  by 
scientific  men,  such  as  have  thoroughly  studied  all  of 
the  principles  of  algebra.  I  have  never,  in  the  course 
of  my  life,  found  occasion  for  extracting  the  cube  root 
once,  for  practical  purposes.  Hence,  the  propriety  of 
excluding  it  from  this  treatise ;  as  likewise  all  of  those 
subdivisions  of  numbers  whose  explanation  depends 
on  algebraic  principles ;  such  as  the  Positions,  Alliga- 
tion, the  Progressions,  Permutation,  etc.,  etc.;  none  of 
which  offer  any  reward  for  the  arduous  labor  lost  in  the 
impossible  task  of  their  attainment  in  arithmetic.  If 


272  RAINEY'S  IMPROVED  ABACUS. 

one-half  the  time  devoted  to  these  principles  in  their 
arbitrary  form  in  arithmetic,  were  devoted  to  the  study 
of  algebra,  the  pupil  would  learn  a  great  portion  of 
that  beautiful  science,  and  thus  secure  the  only  key  to 
the  principles  involved  in  these  rules. 

The  sign  ,J  placed  before  a  number,  indicates  that 
the  square  root  is  to  be  extracted.  By  placing  3,  4, 
5,  etc.,  over  it,  we  understand  that  the  cube,  fourth, 
or  fifth  root  is  to  be  extracted  ;  thus, 

6=r4;  and  4x3=J144. 


SUMMARY   OF   DIRECTIONS. 

I.  Separate  the  number  into  columns  or  periods  of  two  figures 
each,,  by  placing  a  period  (  .  )  over  the  unifs  figure,  and  over 
every  second  figure  from  this  to  the  left,  and  in  decimals,  over 
every  second  figure  toward  the  right. 

II.  Find  the  greatest  square  number  in  the  first  period  to  the 
left,  and  place  the  root,  or  an  equal  factor  of  such  square  number, 
for  the  quotient,  at  the  right  of  the  whole  number:  subtract  the 
square  of  this  root,  or  quotient  figure,  from  the  first  period,  and 
to  the  right  of  the  remainder  bring  down  the  two  figures  of  the 
next  period,  for  a  new  dividend. 

III.  Double  the  root  or  quotient  figure  obtained,  and  place  it 
to  the  left  of  the  new  dividend,  for  a  new  partial  divisor:  ascer- 
tain how  many  times  it  is  contained  in  the  new  dividend,  ex- 
clusive of  the  right-hand  figure  of  such  dividend,  and  place  the 
quotient   to  the  right  of  the    first  quotient  or  root:    place  this 
quotient,  or  second  figure  of  the  root,  likewise  to  the  right  of 
the  partial  divisor,  which  was  used  in  obtaining  it:   multiply 
the  whole  number  thus  found  as  the  divisor,  by  the  figure  thus 
appended,  or  the  last  figure  in  the  root:  subtract  the  product,  and 
bring  down  the  next  period,  for  a  new  dividend. 

IV.  Proceed  with  this  period  as  the  one  preceding,  and  thus 
continue  the  operation,  until  the  roots  of  all  the  periods  are  ex- 
tracted. 

V.  If  there  be  a  remainder  after  all  the  periods  are  thus  used, 
two  ciphers  may  be  added  at  a.  time,  and  the  operation  continued 
to  any  desired  number  of  decimal  places. 

VI.  The  work  is  correct,  if  the  root  multiplied  by  itself,  gives 
a  product  equal  to  the  original  number. 


SQUARE    ROOT.  273 


What  is  the  square  root  of  729? 

It  this  example,  we  make  7  the  first, 
and  29  the  second  period :  2  squared, 
equal  to  4,  makes  the  largest  number 


2)729(37 
4 


that  can   be   extracted  from   7 ;  for  3  I  47^090 
squared,   would  give  9,    a   number  too  |        399 
large.     We  place  this  2  to  the  right, 
subtract  its  square,  leaving  3,  and  bring  0 

down  the  20:  we  now  square  the  2,  placing  the  square 
4,  on  the  left  of  the  new  dividend,  829 ;  we  divide 
the  4  into  the  first  two  figures  of  the  dividend,  32, 
and  find  that  it  would  go  eight  times,  and  conclude 
that  8  must  be  placed  to  the  right  of  the  divisor,  4 ; 
but,  when  we  multiply  the  48  thus  found  by  the  root 
8,  we  find  that  it  makes  384,  a  number  that  cannot 
be  subtracted  from  329.  Hence,  we  conclude  that 
the  partial  divisor,  4,  must  go  into  32  only  seven 
times,  making  allowance  for  the  one  that  will  be  car- 
ried from  the  9  which  is  rejected,  and  place  the  7  in 
in  the  root,  and  also  at  the  right  of  the  divisor  4, 
making  47.  Now,  this  47,  multiplied  by  the  root,  7, 
makes  329,  which  subtracted  leaves  nothing.  We, 
therefore,  conclude  that  the  root  is  27,  and  prove  it 
by  finding  that  27 X27=729.  Again : 

What  is  the  side  of   a  square  field  that  contains 
42025  square  roods  ? 

We  first  divide  this  number  into  periods ;  thus, 

The  first  root  is  2,  and  its  pro- 


2)42025(205 
4 

405)2025 
2025 


duct,  4,  subtracted  leaves  nothing. 

20  being  the  next  period,  we  know 

that  4,  which  is  the  root  doubled, 

would  be  contained  5  times ;  but, 

as  45  multiplied  by  5  could  not  be 

subtracted   from    20,  we  say  that 

this  period  gives  no  root  figure,  and  supply  its  place 

by  a  cipher  in  the  root,  and  also  a  cipher  at  the  right 

of  the  partial  divisor,  4.     Now,  this   partial   divisor, 


0 


274 


RAINEY'S  IMPROVED  ABACUS. 


40,  is  contained  in  the  202,  five  times  :  hence,  \ve 
place  5  in  the  root,  and  5  at  the  right  of  40,  and  mul- 
tiply the  divisor  thus  found  by  the  5,  making  2025. 
Hence,  the  root  is  205.  This  multiplied  by  itself 
gives  the  original  number. 

In  decimals  appended  to  a  whole  number,  the  ex- 
traction is  effected  as  in  whole  numbers.  The  only 
tiling  to  be  observed,  is  to  place  the  decimal  point 
after  the  last  root  figure  in  the  whole  numbers,  and 
all  of  the  remaining  figures  will  be  decimals. 

To  ascertain  the  square  root  of  a  vulgar  fraction,  reduce  ike 
fraction  to  its  lowest  term,  and  extract  the  roots  of  the  nume- 
rator and  denominator,  separately. 


What  is  the  square  root  of  y8 

81  =  9  J9=3 

9)777    7^  ;   now>  ~ 

144=lb 


the  roots  of 


the  fraction,  and  reduce  these  to  the  lowest  term; 

/81  =  9  9  =3 

thus,  -     —  ;    and  —    —  Ans. 
7144=12'          12=4 

A   number  whose   root   cannot  be  exactly  ascer- 
tained, is  called  a  surd. 


CURRENCY. 

Value  of  foreign  Gold  and  Silver  Coins,  according  to  Custom- 
House  usage. 


Guinea,  English,  gold,  $5.00 

Crown,         "        silver,  1.12 

Shilling,       «            «  .23 

Bank  token,  English,  silver,  .25 

Florin,  of  Basle,  silver,  .41 

Moidore,  Brazil,  gold,  4.80 

Livre,  of  Catalonia,  silver,  .53K 

Florence  Livre,  silver,  .15 

Louis  d'or,  French,  gold,  4.56 

Crown.            "         silver,  1.06 

40  Francs,       "        gold,  7.66 

5  Francs,        "        silver,  .93 

Geneva  Livre,  silver,  .21 

10  Thalers,  German,  gold,  7.80 

10  Pauls,  Italy,  silver,  .97 

Jamaica  Pound,  nominal,  3.00 


Leghorn  Dollar,  silver,  $0.90 

Sen  do,  of  Malta,     "  .40 

Doubloon,  of  Mexico,  gold,  15.60 

Livre,  of  Neufchatel,  silver,  .26 % 

Half  Joe,  Portugal,  gold,  8.53 

Florin,  Prussia,  silver,  .22% 

Imperial,  Russia,  gold,  7.83. 

Rix  Dollar,  Rhenish,  silver,  .60% 

"        "        Saxony,        "  .69 

Pistole,  Spanish,  gold,  3.97 

Rial,             «         silver,  .12>£ 

Cross  Pistareen,       "  .16 

Other  Pistareens,      "  18 

Swiss  Livre,              "  .27 

Crown,  of  Tuscany,  silver,  1.05 

Piaster,  Turkish,          "  .05 


FOREIGN    COINS    AND    MONIES    OF  ACCOUNT.      275 
Monies  oj  Account  and  Coins^made  current  by  act  of  Congress.* 


Pound  Sterling,  G,  Britain,  $4.84 
Do.,  Canada  and  N.  Sc«.,       4.00 
Do.,  N.  Bran,  and  N.  Found.,  4.00 
Franc,  of  France  and  Belgium,  .186 
Livre  Tournois,  France,             ..185 
Florin,  Netherlands,                    .40 
Do.,  Southern  Ger.  States,        .40 
Guilder,  of  Netherlands,            .40 
Real  Vellon,  Spain,                    .05 
Do.,  Plate,        «                         .10 
Milree,  of  Portugal,                  1.12 
Milree.  of  Azores,                        .83% 
Marc  Banco  of  Hamburg,           .35 
Thaler,  or  Rix  Dollar,  Prussia, 
and  Nor.  Ger.  States,              .69 

Rix  Dollar  of  Bremen,          $0.78% 
Specie  Dollar  of  Denmark,      1.05 
Do..,  Sweden  and  Norway,      1.06 
Rouble,  Russia,  silver,               .75 
Florin,  of  Austria,                      .485 
Lira  or  Lambardo,  Venetian 
kingdom,                                    .16 
Lira,  of  Tuscany,            .            .16 
Lira,  of  Sardinia,                         .186 
Ducat,  Naples,                            .80 
Ounce,  of  Sicily,                      2.40 
Livres,  Leghorn,                          .16 
Tael,  of  China,                          1.48 
Rupee,  Company,                        .445 
Pagoda,  India,                          1.84 

Foreign  Monies  of  Account,  giving  the  value  of  the  unit,  accord- 
ing to  custom,  in  dollars  and  cents.^ 

Brazil.— 1000  Rees  =-1  Milree  =  in  Federal  money  to  $0.828 

The  silver  coin,  1200  Rees  = 994 

Bremen. — 5  Schwares  =1  Grote;    72  Grotes  =1  Rix 

Dollar,  silver, .787 

Belgium. — 100  cents  =1  Guilder  or  Florin;  1  Guilder, 

(silver),. ,. . 40 

Bencoolen. — 8  Satellers— 1  Soocoo;  4  Soocoos=l  dol~ 

lar  or  rial, ....  — 1.10 

Austria. — 60  Kreutzers  —1  Florin;  1  Florin,  silver  =»      .485 
British  India. — 12  Pice  =1   Anna;  16  Annas  =1  Co 

Rupee,  silver, ....... .  — . . .      .445 

In  Bengal,  Madras,  and  Bombay,  the  current  silver 

Rupee  = 444 

Buenos  Ayres. — 8  Rials  =1  dollar,  common  currency, 

(fluctuating), 93 

Canton. — 10  Cash  —  1  Candarine;    10  Candarines  =1 

Mace;  10  Mace=l  Tael, ]  .48 

(The  Cash,  composed  of  Copper  and  lead,  is  said  to 

be  the  only  money  coined  by  the  Chinese.) 
Cape  of  Good  Hope. — 6  Stivers  =1  Schilling;  8  sen-  =» 

1  Rix  dollar, 313 

Ceylon.— 4  Pice=l  Fanam;  12  Fanarns  =  1  Rix  dol.,       40 

Cuba. — 8  Rials,  plate,  =1  dollar;  1  dollar, 1.00 

Columbia,  Ecuador,  Venezuela,  and  New  Grenada. — 8 

Rials  =1  dollar;  1  dollar,  fluctuating, 1.00 

*  Monies  of  Account  are  not  represented  by  coin,  and  are  used  for  fa- 
cility in  reckoning,  only,  as  mills  in  our  country.  To  ve/ify  the  tables 
above,  see  Laws  of  the  United  States. 

.tSee  Encyclopaedia  JBritaunica,  and  McCulloctfs  Commercial  Dictionary 


276  RAINEY'S  IMPROVED  ABACUS. 

Chili.— 8  Rials  =1  dollar;  1  dollar,  silver,  = $1.00 

Denmark. — 12  Pfennigs  =1  skilling;  16  Sk.  =1  Marc; 

6  Marcs  =1  Rigsbank,  or  1  Rix  dollar,  silver, 52 

Egypt-—*  Aspers  =1  Para;  40  Paras  =1  Piaster,  sil.,       .048 
Hamburg.— 12  Pfenings  =1  Schilling  or  Sol;  16  Schil. 
=1  Marc  Lubs;*  3  Marcs  =1  Rix  dollar,  Current 

Marc,  silver, 28 

Marc  Banco, 35 

Holland.— 100  Cents  =1  Florin,  or  Guilder:  1  Florin, 

silver, 40 

Greece. — 100  Lepta  =1  Drachme;  1  drachme,  silver,. .      .166 
Great  Britain  and  France. — See  tables  above. 
Japan. — 10  Candarines  ==1  Mace;  10  Mace  ==1  Tael,        .75 
Malta.— -20  Granif  =1  Taro;  12  Tari  =1   Scudo;  2>£ 

Scudi  =1  Pezza, 1.00 

Java.-— 100  Cents  =1  Florin;  1  Florin,  as  in  Nether- 
lands,  40 

Mauritius.— In  accounts  of  state,  100  Cts.  =1  dol.  =       .968 
Manilla.— -34  Maravedies  =1  Rial;  8  Rials  =1   dollar, 

Spanish, 1.00 

Milan.— 12  Denari  =1  soldo;  20  Soldi  =1  Lira, 20 

Mexico.— -8  Rials  =1  dollar,  1  dollar, 1.00 

Montevideo.— 100  Centesimi  =1  Rial;  8  Rial  =1  dol.,      .833 
Naples.— 10  Grani  =1   Carlino;  10  Carlini  =1  Ducat, 

silver, 80 

Netherlands. — Throughout  the  whole  kingdom,  ac- 
counts are  kept  in  Florins  or  Guilders,  and  cents,  as 
per  law  of  1815.  See  Holland. 

New  South  Wales. — Accounts  are  kept  in  Sterling 
Money,  only. 

Norway 4— 120  Skillings  ==1  Rix  dollar,  silver, 1.06 

Papal  States.— 10   Bajocchi  =1   Paolo;  10  Paoli  =1 

Scudo  or  Crown, 1.00 

Peru.— 8  Rials  =1  dollar,  silver, 1.00 

Portugal— 400  Rees  =1  Cruzado;^  1000  Rees  =--1  Mil- 
ree  or  Crown, 1.12 

*  Lubs  indicates  that  it  is  the  money  of  the  city  Lubec;  the  common  coin 
is  the  marc  currency;  the  marc  banco  represents  the  certificates  of  deposit 
of  bullion,  jewelry,  etc.,  in  the  bank  of  Hamburg.  Invoices  and  accounts 
are  frequently  made  in  Flemish  pounds,  shillings,  and  pence,  which  are 
subdivided  as  sterling  money.  The  Flemish  pound  is  equal  ta7%  marcs 
banco. 

t  Grani  is  the  plural  of  grano;  tari,  plural  of  taro;  scudi,  of  scudo;  lire, 
of  lira;  pezze  of  pezza;  soldi,  of  soldo;  carlini,+f>f  carlinoj  bajocchi,  of 
bajoccho;  and  paoli,  of  paolo. 

t  Norway  has  no  gold  coin  of  her  owu. 

§  Cruzadi  is  plural  of  cruzado;  groschcn,  of  grosch;  centesimi  of  ctnti- 
lima;  lire  piecolet  of  lira  pictola;  soldi  di  pezza>  of  soldo  di  pezza* 


JEWISH    WEIGHTS    AND    MEASURES.  277 

Prussia. — 12  Pfennigs  =1  Grosch,  silver;  30  Groschen 
=1  Thaler,  or  dollar, $0.69 

Russia.— 100  Copecks  =1  Rouble,  silver, 78 

Accounts  were  kept  in  paper  Roubles  previous  to 
1840,  3j^  of  which  were  equal  to  1  silver  Rouble. 

Sardinia. — 100  Centesimi  =1  Rira;  1  Lira=l  Franc, 
French, 186 

Sweden.— 12  Rundstycks -=1  Skilling;  48  Skilling  =1 
Rix  dollar,  specie, 1.06 

Sicily.— 20  Grani=l  Taro;  30  Tari==l  Oncia,  gold,       2.40 

Spain. — 2  Maravedies=l  Quinto;  16  Quintos=l  Rial 
of  old  plate:*  20  Rials  vellon  =1  Span,  dollar, 1.00 

St.  Domingo.— 100  Centimes  =1  dollar:  1  dollar, 33)£ 

Tuscany. — 12  Denari  di  Pezza  =1  Soldo  di  Pezza;  2 
Soldi  di  Pezza  =1  Pezza  of  8  Rials;  1  Pezza,  silver,  .90 

Turkey.— 3  Aspers  =1  Para;  40  Paras  =1  Piaster, 
varying, 05 

Venice.— 100  centesimi  =1  Lira;  1  Lira  =1  Franc,  Fr.,     .186 
Accounts  were  once  kept  in  ducats,  lire,  etc.     12 
Denari  =1   Soldo;    20  Soldi  =1    Lira  Piccola;    6j. 
Lire  piccole  =1  Ducat  current;  8  Lire  pic.  =1  Ducat 
effective;  the  Lira  piccola  is  worth,    096 

West  Indies,  British. — Pounds,  shillings,  pence,  etc.,  as 
in  England;  the  value  varies  in  the  different  islands, 
and  is  in  all  of  them  below  that  of  England. 

Jewish  or  Scripture,  Standard  Weights  and  Measures.^ 

WEIGHTS  OF  MONEY. — 60  Shekels  =1  Maneh ;  50  Maneh 
=1  Talent;  or  113  Ibs.,  10  oz.,  1  clwt.,  10  grs.,  Troy. 

LONG  MEASURE. — 4  Digits  ==1  Palm:  3  Palms  ^=1  Span;  2 
Spans  =1  Cubit;  4  Cubits  =1  Fathom;  2  Fathoms  =1  Ara- 
bian Pole;  10  Poles  =1  Schoenus,  which  is  the  measuring 
line,  and  is  equal  to  144  feet,  11  inches. 

ITINERARY  MEASURE. — 400  Cubits  =1  Stadium;  5  Stadia 
=1  Sabbath-day's  Journey;  10  Stadia  =1  Eastern  Mile;  3 
Eastern  Miles  =1  Parasang;  8  Parasangs  =1  Day's  Journey, 
or  33|  English  Miles. 

DRY  MEASURE. — 20  Grachal  =1  Cab;  1  A  Cabs=l  Gomor; 
3>£  Gomor  =1  Seah;  3  Seahs  =1  Ephah;  5  Ephahs  =1  Le- 
leeh;  2  Leteeh  =1  Comer,  or  2  Bushels,  1  pint,  English. 

LIQUID  MEASURE. — lj/g  Caph  ==1  Log;  4  Logs  =1  Cab; 
3  Cabs  =1  Hin;  2  Hins  =1  Seah;  3  Seahs  =1  Bath  or 

*  Although  rial  of  old  plate  is  not  a  coin,  yet  it  is  the  denomination  in 
which  exchanges  and  invoices  are  reckoned. 
ISce  Kelly's  Universal  Cambist. 


278 


RA1NEY  S    IMPROVED    ABACUS. 


Ephah;  10  Ephah  s  =1   Chomer,  Homer,  or    Corus,   which 
equals  75  gallons,  5  pints,  English  Measure. 


1  Talents=113  Ibs.,  10  oz.,  1 

dwt.,  10  grs;  or  655714  grs. 

1  Maneh  ==13114.28  grs.;  or, 

27.3214  oz.;  or, 

2.27678  Ibs.  T. 

1  Shekel  =     218.57133  grs. 

1  Schoenus=145ftll  in.;  or, 
1          "        =1751.  inches. 
1  Pole          =  175.1     " 
1  Fathom    =     87.55  ",      or, 

7  ft.  3%  inches. 
1  Cubit    =~  21. 8875  inches. 
1  Span     — 10.9437      « 
1  Palm      =    3.6477      " 
1  Digit     —      .9119      « 

1  Day's  Journey  =33i   Eng- 
lish miles;  or  58374.3216  yds. 
1  Parasang  =7296.7902       « 


1  Eas.  Mile  =2432.2634  yards. 
1  Stadium  —  243.2263     « 
1  Sab.  D.  J  .=1216.1315    « 

1  Comer    =2  bu.  1  pt.,  Eng.; 
or  16.125  galls.;  or  129.  pts. 
1  Leteeh     =64.5         pints. 
1  Ephah     =12.9 
1  Seah        —  4.3 
1  Gomor    =  1.29  « 

1  Cab         =    .7166       « 
1  Grachal  =     .0358      « 

1  Chomer  =15  galls.,  5  pts., 

English;  or  605  pints. 
1  Ephah  =60.5     pints. 
1  Seah     =21.166 
1  Kin      =10.083 
1  Cab      =  3.361 
1  Log      =     .8402 
1  Caph    =     .6301 


1  Talent,  silver  =$1589.61;  of  gold,  =$25415.27.  1  Ma- 
neh,  silver  =$31.79;  gold,  =$508.22.  1  Shekel,  =$0.529; 
gold,  =$847;  all  24  carets  fine,  allowing  no  alloy. 

Time  Table,  for  Banking  and  Equation,  giving  the  number  of 
days  from  any  given  date  in  one  month,  to  the  same  date  in 
any  other  month. 


A.D. 

1849. 

a 

53 

1 

9 

< 

ri 
~, 

a> 

c 

o 

1   II 

III 

Jan. 

365 

31 

59 

90 

120 

151 

181 

212  243 

273 

304  334 

Feb. 

334 

365 

28 

59 

89 

120 

150 

181  212 

242 

272  303 

Mar. 

306 

337 

365 

31 

61 

91 

122 

153  184 

214 

245  275 

Ap'l. 

275 

306 

334 

365 

30 

61 

91 

122  153 

183 

214  244 

May. 

245 

276 

304 

335 

365 

31 

61 

92  123 

153 

184  214 

June. 

214 

245 

273  304  1334  365 

30 

61  i  92 

122 

153  183 

July. 

184 

215 

243  |274  |3Q4 

335 

365 

31  !  62 

92 

123  153 

Aug. 

153 

184 

212  243  273 

304  334 

365     31 

61 

92  122 

Sept. 

122 

153 

181  212  -242 

273  303 

334  365 

30 

61     91 

Oct. 

92 

123 

151  182  212  243  273 

304  335 

365 

31     61 

Nov. 
Dec. 

61 
31 

92 
62 

120  151  181 
90  11211151 

212  242  273  304  334  365     30 
182  i!212  1243  274  1304  1335  365 

TABLE    OF    DIAMETERS    AND    AREAS. 


279 


The  number  of  days  expiring  between  any  two  periods  may  be  very 
easily  ascertained  in  the  foregoing  table,  by  ascertaining  the  time  between 
the  first  and  second  dates,  and  adding  or  subtracting  the  overplus,  or 
deficit,  minus  1.  For  example: 

How  long  does  a  note  run,  dated  January  4,  and  payable  December  14? 

In  the  table  above,  from  Jan.  4,  to  Dec.  4,  is  334  days;  and  9  days  more, 
added,  excluding  the  latter  date,  make  343  days,  the  time  that  the  note 
runs. 

How  long  does  a  note  run  from  December  4,  to  January  41 

In  the  left-hand  column  we  find  December,  and  opposite  it,  under  the 
head  Jan.,  we  find  31  days,  the  time.  The  month  of  the  first  date  must  be 
sought  in  the  column  at  the  left. 

For  leap-year,  one  must  be  added  to  the  number  of  days,  when  the  month 
of  February  comes  within  the  two  dates. 


TABLE 


Of  Diameters  and  Areas  of  Circles. 


Dia. 

Area. 

Dia. 

Area. 

Dia. 

Area. 

Dia. 

Area. 

Dia. 

Area. 

lin. 

.7854 

IOK 

82.516 

19  K 

298.648 

28% 

649.182 

38 

1134.11 

IX 

1.2271 

10  ^ 

86.590 

19% 

306.355 

29 

660.521 

38X 

1149.08 

\yz 

1.7671 

10?£ 

90.762 

20 

314.160 

29^ 

671.958 

38^ 

1164.15 

1% 

2.4052 

11 

95.035 

20^ 

322.063 

29  yz 

683.494 

38% 

1179.32 

3.1416 

ivx 

99.402 

20  y* 

330.064 

29% 

695,128 

39 

1194.59 

2K 

3.9760 

11  y' 

103.869 

20% 

338.163 

30 

706.860 

39K 

1209.95 

2>£ 

4.9087 

ft% 

108.434 

21 

346.361 

30% 

718.690 

39^ 

1225.42 

5.9395 

12 

113.097 

21K 

35-1.657 

30^ 

730.618 

39% 

1240.98 

3  * 

7.0686 

12^ 

117.859 

21  yz 

363.051 

30% 

744.644 

40 

1256.64 

3K 

8.2957 

™% 

122.718 

aig 

371.543 

31 

754.769 

40^ 

1272.39 

3>£ 

9.6211 

!2% 

127.676 

22 

380.133 

31K 

766.992 

40  K 

1288.25 

3% 

11.044 

13 

132.732 

22K 

388.822 

31X 

779.313 

40% 

1304.20 

4 

12.566 

!3K 

137.886 

&X 

397.608 

31% 

791.732 

41 

1320.25 

4K 

14.186 

13  y* 

143.139 

22% 

406.193 

32 

804.249 

41X 

1336.40 

4% 

15.904 

13% 

148.489 

23 

415.476 

32^ 

816.865 

41  K 

1352.65 

4% 

17.720 

14 

153.938 

23>4: 

424.557 

32  yz 

829.578 

41% 

1369.00 

5 

19.635 

14^ 

159.485 

23  yz 

433.731 

32% 

842.390 

42 

1385.44 

5% 

21.647 

!4^ 

165.130 

23% 

443.004 

33 

855.30 

42X 

1401.98 

5>a 

23.758 

m 

170.873 

24 

452.390 

33^ 

868.30 

42^ 

1418.62 

4! 

25.967 

15 

176.715 

24^ 

#1.864 

33^ 

881.41 

42% 

1436.36 

6 

28.274 

15^ 

182.654 

24K 

471.436 

33% 

894.61 

43 

1452.20 

6X 

30.679 

«% 

188.692 

24% 

481.106 

34 

907.92 

43X 

1469.13 

6K 

33.183 

U% 

194.828 

25 

490.875 

34X 

921.32 

43  yz 

1486.17 

6% 

35.784 

16 

201.062 

25K 

500.741 

**y* 

934.82 

43% 

1503.30 

7 

38.484 

16>^  i  207.394 

25  K 

510.706 

34% 

948.41 

44 

1520.53 

7K 

41.282 

16  >/ 

213.825 

25% 

520.769 

35 

962.11 

44X 

1537.86 

7K 

44.178 

16% 

220.353 

26 

530.930 

35^ 

975.90 

44  yz 

1555.28 

TX 

47.173 

17 

2*5.980 

26^ 

541.189 

85« 

989.80 

44% 

1572.81 

8 

50.265 

ft* 

233.705 

26K 

551.547 

35% 

1003.70 

45 

1590.43 

8K 

53.456 

17$ 

240.528 

26% 

5ty.00'3 

36 

1017.87 

45K 

1608.15 

8^ 

56.745 

ITX 

247.450 

27 

572.556 

36M 

1032.06 

45}^ 

1625.97 

8% 

60.132 

18 

254.469 

27X 

5S3.208 

36  >/ 

1046.30 

45% 

1643.89 

9 

63.617 

18X 

261.587 

27  >^ 

593.958 

36% 

1060.73 

46 

1661.90 

9X 

67.200 

18K 

268.803 

27% 

604.807 

37 

1075.21 

46X 

1680.01 

9K 

70.882 

18% 

276.117 

28 

615.753 

37^ 

1089.79 

46  >^ 

1698.23 

9% 

74.662 

19 

383.888 

28^ 

626.798 

37  >£ 

1104.46 

46% 

1716.54 

10 

78.540 

19X 

291.039 

28^ 

637.941 

37% 

1119.24 

47 

1734.94 

280 


RAINEY'S  IMPROVED  ABACUS. 


TAB  LE—  C&nt  inued. 


Dia. 

Area. 

Dia. 

Area. 

Dia. 

Area. 

Dia. 

Area. 

Lia. 

Area. 

47% 

1753.45 

57% 

2619.35 

68K 

3658.44 

85  yz 

5741.47 

9    8 

73.391 

47% 

1772.05 

58 

2642.08 

68% 

8685.29 

86 

5808.81 

9    9 

74.662 

47% 

1790.76 

58% 

2664.91 

68% 

3712.24 

86K 

5876.55 

9  10 

75.943 

48 

1809.56 

58  % 

2637.83 

69 

3739.28 

87 

5944.69 

9  11 

77.236 

48% 

1828.46 

58% 

2710.85 

69% 

3766.43 

87>ax 

6013.21 

10 

78.540 

48% 

1847.45 

59 

2733.97 

09% 

3-793.67 

88 

6082.13 

10    1 

79.854 

48% 

1868.55 

59^ 

2757.19 

69% 

3821.02 

88  K 

6151.44 

10    2 

81.179 

49 

1885.74 

59% 

2780.51 

70 

3848.46 

89 

6221.15 

10    3 

82.516 

49K 

1905.03 

69% 

2803.92 

1Q14 

3875.99 

89% 

6291.25 

10    4 

83.862 

49% 

:  1924.42 

60 

2827.44 

70% 

3903.63 

90 

6361.74 

10    5 

85.221 

49% 

1943.91 

60^ 

2S51.05 

70% 

3931.36 

90% 

6432.62 

10    6 

86.590 

60 

1963.50 

60% 

2874.76 

71 

3959.20 

91 

6503.89 

10    7 

87.969 

50% 

1983.18 

60% 

2898.59 

71# 

3987.13 

91  % 

6573.56 

10    8 

89.360 

50  % 

2002.96 

61 

2922.47 

71% 

4015.16 

92 

6647.62 

10    9 

90.762 

50% 

2022.84 

61M 

2946.47 

71% 

4043.28 

92% 

6720.07 

10  10 

92.174 

51 

2042.82 

6i  % 

2970.57 

72 

4071.51 

93 

6792.92 

10  11 

93.598 

51% 

2062.90 

61% 

2994.77 

72% 

4128.25 

93% 

6866.16 

11 

95.033 

51% 

2083.07 

62 

3019.07 

73 

4185.39 

94 

6939.79 

11    1 

96.478 

51% 

2103.35 

62)^ 

3043.47 

73% 

4242.92f 

94% 

7013.81 

11     2 

97.934 

52 

2123.72 

62% 

3067.96 

74 

4300.85 

95 

7088.23 

11     3 

99.402 

52K 

2144.19 

62% 

3092.56 

74K 

4359.1'') 

95% 

7163.04 

11     4 

100.879 

52% 

2164.75 

63 

3117.25 

75 

4417.87 

ft.  in. 

feet. 

11    5 

102.368 

52% 

2185.42 

63X 

3142.04 

75  y* 

4476.97 

8 

50.265 

11    6 

103.869 

53 

2206.18 

63% 

31G6.92 

76 

4536.47 

8-  1 

51.317 

11    7 

105.379 

53% 

2227.05 

63% 

3191.91' 

76« 

4596.35 

8    2 

52.381 

11    8 

106.901 

53% 

2248.01 

64 

3216.99 

77 

4656.63 

8    3 

53.456 

11    9 

108.434 

53^ 

2269.06 

64# 

3242.17 

77% 

4717.30 

8    4 

54.541 

11  10 

109.977 

54 

2290.22 

64% 

3267.46 

78 

4778.37 

8    5 

55.637 

11  11 

111.531 

54% 

2311.48 

64% 

3292.83 

78>£ 

4839.83 

8    6 

56.745 

12 

113.097 

54% 

2332.83 

65 

3318.31 

79 

4901.68 

8    7 

57.862 

13 

132.732 

54% 

2354.28 

65^ 

3343.88 

79% 

4963.92 

8    8 

58.992 

14 

153.938 

55 

2375.83 

65  % 

3369.56 

80 

5026.56 

8    9 

60.132 

15 

176.715 

55K 

2397.48 

65% 

3395.33 

80  % 

5089.58 

8  10 

61.282 

16 

201.062 

55% 

2419.22 

66 

3421.20 

81 

5153.00 

8  11 

62.444 

17 

226.980 

55% 

2441.07 

66K 

3447.16 

8L& 

5216.83 

9 

63.617 

18 

254.469 

56 

2463.0r 

66  % 

3473.23 

82 

5281.02 

9    1 

64.800 

19 

283.529 

56K 

2485.05  ; 

66% 

3499.39 

82  y, 

5345.0-3 

9    2 

65.995 

20 

314.160 

56% 

2507.19 

67 

3525.66 

8S 

5410.62 

9    3 

67.200 

21 

346.361 

56% 

2529.42 

67^ 

3552.01 

83^ 

5476.00 

9    4 

68.416 

22 

380.133 

57 

2551.76 

67% 

3578.47  : 

84 

5541.78 

9    5 

69.644 

23 

415.476 

57K 

2574.19 

67% 

3605.03 

84K 

5607.90 

9    6!    70.833 

24 

452.390 

57% 

2596.72 

68 

3631.68 

85 

5674.51 

9    71    72.130 

25 

490.87* 

Weight  of  a  Lineal  Foot  of  Square  Rolled  Iron,  in  lbs.,from 
to  12  inches  square. 


Size, 

Wei't, 

Size,    Wei't, 

Size, 

Wei't, 

Size, 

Wei't, 

Size, 

Wei't, 

in  in. 

inlbs. 

in  in. 

in  Ibs. 

in  in. 

in  Ihs. 

in  in. 

inlbs. 

in  in. 

in  Ibs. 

K 

.211 

% 

2.588 

1£ 

7.604 

2% 

15.263 

2% 

25.560 

% 

.475 

1 

3.380 

1% 

8.926 

2% 

17.112 

2% 

27.939 

.845 

1% 

4.278 

1% 

10.352 

2% 

19.066 

3 

30.416 

i£ 

1.320 

1% 

5.280 

1% 

11.883 

2% 

21.120 

3% 

33.010 

% 

1.901 

lg 

6.390 

2 

13.520 

2% 

33.292 

3K 

35.704 

WEIGHT  OF  DIFFERENT  BODIES  OF  IRON. 


281 


TABLE—  Continued. 


Size, 
in  in. 

Wei't, 
in  Ibs. 

Size,, 
in  in. 

Wei't, 
in  Ibs. 

Size, 
in  in. 

Wei't, 
in  Ibs. 

Size, 
in  in. 

Wei't, 
in  Ibs. 

Size, 
in  in. 

Wei't, 
in  Ibs. 

3K 

86.503 

4% 

72.305 

5^ 

111.756 

7& 

203.024 

10 

337.920 

B£ 

41.408 

4% 

76.264 

5K 

116.671 

8 

216.336 

10K 

355.136 

3% 

44.418 

4% 

80.333 

6 

121.664 

8K 

230.068 

10  K 

372.672 

3% 

47.534 

5 

84.480 

6K 

132.040 

8K 

244.220 

10% 

390.628 

3% 

50.756 

5K 

88.784 

6K 

142.816 

8% 

258.800 

11 

408.960 

4 

54.084 

5K 

93.168 

6% 

154.012 

9 

273.792 

UK 

427.812 

4K 

57.517 

5% 

97.657 

7 

165.632 

9K 

289.220 

UK 

447.024 

4K 

61.055 

5K 

102.240 

7K 

177.672 

9K 

305.056 

11% 

466.684 

4K 

64.700 

5% 

106.953 

7K 

190.136 

9& 

321.332 

12 

486.656 

4K 

68.448 

Weight  of  Round  Rolled  Iron,  1  foot  long,  and  from  %  to  12 
inches  in  diameter. 


- 


Dia., 

Wei't, 

Uia-, 

Wei't, 

Uia., 

Wei't, 

Dia., 

Wei't, 

Dia., 

Wei't, 

in  in. 

in  Ibs. 

in  in. 

in  Ibs. 

in  in. 

in  Ibs. 

in  in. 

in  Ibs. 

in  in. 

in  VSs. 

K 

.165 

2K 

11.988 

8# 

39.864 

5% 

84.001 

8% 

203.269 

% 

.373 

2K 

13.440 

4 

42.464 

5?4 

87.776 

9 

215.040 

% 

.663 

a% 

14.975 

4X 

45.174 

5% 

91.634 

9K 

227.152 

% 

1.043 

2K 

16.688 

4V 

47.952 

6 

95.552 

9K 

239.600 

% 

1.493 

a* 

18.293 

4?< 

50.815 

ex 

103.704 

93^ 

252.376 

% 

2.032 

2% 

20.076 

4K 

53.760 

6# 

112.160 

10 

266.288 

1 

2.654 

^ 

21.944 

4% 

56.788 

6%r 

120.960 

10K 

278.924 

IK 

3.360 

3 

23.888 

4# 

69.900 

7 

130.048 

10K 

292.688 

IK 

4.172 

*K 

-25.926 

4% 

63.094 

?« 

139.544 

10% 

306.800 

IK 

5.019 

3X 

28.040 

5 

66.752 

»K 

149.328 

11 

321.216 

IK 

5.972 

3% 

30.240 

5K 

69.731 

7% 

159.456 

UK 

336.004 

1% 

7.010 

3K 

32.512 

5# 

73.172 

8 

169.856 

UK 

351.104 

IK 

8.128 

3% 

34.886 

5'< 

76.700 

8* 

180.696 

u% 

3b6.536 

IK 

9.333 

8* 

37.332 

5^ 

80.304 

8^ 

191.808 

12 

382.208 

2 

10.616 

Weight,  in  Ibs.,  of  different  bodies  of  Cast  Iron,  1  foot  in  length, 
and  from  %  to  12  inches  diameter  or  side. 


Side, 
ordi. 

Squa. 

Hex  a- 
gon. 

Octa- 
gon. 

Circle. 

Side, 
or  di. 

Squa. 

Hex  a- 
gon. 

Octa- 
gon. 

Circle. 

iru-b. 

inch. 

y, 

.781 

.675 

.650 

.612 

3K 

33.009 

28.565 

27.475 

25.921 

X 

1.75(5 

1.528 

1.471 

1.387 

8« 

38.281 

33.131 

31.818 

30.065 

1 

3.125 

2.703 

2.603 

2.454 

3% 

43.943 

38.031 

36.581 

31.512 

IV 

4.881 

4.225       4.065 

3.854 

4 

50.000 

43.271 

41.621 

39.268 

1U 

7.031 

6.085  ;     5.856 

5.521 

4X 

56.443 

48.353 

46.990 

44.331 

1« 

9.568 

8.281  '     7.971 

7.515 

*K 

63,281 

5-1.768 

52.681 

49.700 

2 

12.520 

10.815,   10.412 

9.815 

4fc 

70.506 

61.021 

58.696 

55.375 

2V 

15.818 

13.990  i   13.168 

12.425 

5 

78.125 

67.515 

65.040 

61.359 

2^ 

19.531 

16.900,   16.256 

15.337 

5J4 

86.131 

74.549 

71.701 

.67.709 

2^ 

23.631 

20.450     19.671 

18.559 

5V, 

94.531 

81.815 

78.696 

74.243 

3 

28.125 

24.340     23.412 

22.087 

&% 

103.318 

89.421 

86.015 

81.126 

282 


RAINEY  S    IMPROVED    ABACUS. 


TABLE—  Continued. 


Side, 
or  di. 

Squa. 

Hexa- 
gon. 

Octa- 
gon. 

Circle. 

Side, 
or  di. 

Squa. 

Hexa- 
gon. 

Octa- 
gon. 

Circle. 

inch. 

inch. 

6 

112.500 

97.368 

93.656 

88.354 

9K 

266.781 

231.418 

222.600 

210.800 

6K 

122.058 

105.640 

101.621 

95.871 

9>£ 

282.031 

244.100 

234.793 

221.506 

sy* 

132.031 

114.271 

109.948  1  103.696 

&2 

296.968 

257.105 

247.315 

233.318 

6% 

142.381 

123.231 

118.534 

111.825 

10 

3-12.500 

270.471 

260.163 

245.437 

7 

153.125 

132.528 

127.478 

120.372 

10K 

328.318 

284.159 

273.341 

257.859 

7K 

161.256 

142.162  i  136.743 

128.986 

10^ 

344.531 

298.193 

286.828 

270.593 

175.781 

152.037  !  146.337 

138.056 

IOM 

351.131 

312.559 

300.645 

283.633 

7% 

187.693 

162.449  156.259 

147.415 

11 

378.125 

327.268 

314.796 

296.978 

8 

200.000 

173.099 

166.503 

157.078 

11  ^ 

393.216 

342.315 

329.268 

310.631 

w 

212.693 

184.087  177.071 

167.049 

U>£ 

410.281 

357.693 

344.062 

324.587 

*y2 

225.781 

195.412  187.365 

177.328 

HM 

429.023 

373.325 

359.187 

338.856 

8% 

239.256 

207.078  199.127 

187.912 

12 

450.000 

389.475 

374.613 

353.428 

9 

253.125 

219.078  210.721 

199.203 

i 

Weight  of  a  lineal  foot  of  Flat  Bar  Iron,  in  lbs.,from  %  to 
inches  in  width,  and  from  %  to  5  indies  in  thickness. 


c 

c. 

i 

« 

« 

J 

c  I    c 

»* 

te 

c 

g 

c 

.S 

^ 

g 

e 

c 

c 

a 

fl 

5J  I 

a 

c 

•2 

5 

c 

'  .S 

[c 

i 

£_^ 

g 

i 

g 

g" 

1 

\  ^ 
H 

.j 

£ 

H 

g 

£ 

_H 

g 

** 

TJ 

0.211 

5,/ 

2.375 

% 

4.435 

1  S'8 

9.610 

1% 

12.673 

K 

0.422 

% 

2.850 

1  * 

5.069 

/8 

^ 

0.792 

2Ji 

3>8 

0.89S 

?-i 

0.634 

/8 

3.326 

ji/ 

5.703 

34 

1.584 

K 

1.795 

?8 

>a' 

0.264 

1 

3.802 

IK 

6.337 

2.376 

x  8 

2.693 

0.528 

IK 

3s 

0.528 

6.970 

3^ 

3.168 

3.591 

X 

0.792 

K 

1.056 

[5^1     1^ 

0.686 

/8 

3.960 

1:    /8 

4.488 

1;, 

1.056 

X 

1.584 

K 

1.372 

;-i 

4.752 

x4 

5.386 

& 

X 

0.316 

2.112 

% 

2.059 

js 

5.544 

/8 

6.283 

X 

0.633 

/8 

2.640 

i/ 

2.746 

1 

6.336 

1 

7.181 

0.950 

k 

3.168 

% 

3.432 

11B 

7.123 

1>8 

8.079 

V 

1.265 

3.696 

x^ 

4.119 

IK 

7.921 

8.977 

^ 

1.584 

[X 

4.224 

T/ 

4.8051 

1% 

8.713  j 

1/8 

9.874 

7'8 

l,g 

0.3'39 

MX 

4.752 

\ 

5.492 

1  1'^ 

9.505 

lx/ 

10.772 

X 

0.738 

?:s 

x 

0.580 

6.178  ; 

]  sj 

10.297 

1/8 

11  670 

X 

1.108 

1.161 

IK 

6.864 

1  3'i 

11.089; 

1^4 

12.507 

1.477 

•£ 

1.742 

l/'s 

7.551   2 

38 

0.845  I 

1% 

13.465 

fa 

1.846 

K' 

2.325 

1  1/ 

8.237 

3i 

1.689! 

2 

14.362 

m 

2.217 

2.904 

•X 

// 

0.739 

X 

2.534 

-  1  i 

"K 

0.950 

1 

]& 

0.422 

3i 

3.484 

3i 

1.479 

X 

3.379  ! 

K 

1.900 

14 

0.845 

% 

4.065 

^ 

2.218 

4.224  i 

% 

2.851 

% 

1.267 

4.646 

K 

2.957  ! 

x4 

5.069 

^2 

3.802 

1.690 

i/ 

5.227 

* 

3.696  , 

J.g 

5.914  | 

% 

4.752 

/8 

2.112 

ik 

5.808 

4.435; 

[ 

6.758 

% 

5.703 

;!.i 

2.534 

IK  x 

0.633 

7/8 

5.178 

(X 

7.604 

% 

6.653 

7/ 

2.956 

K 

1.266 

1 

5.914 

8.448 

1 

7.604 

1  "o 

i,/ 

0.475 

1.900 

6.653 

[% 

9.294 

IX 

8.554 

K 

0.950 

i> 

2.535 

H^ 

7.393 

IX  10-138 

IK 

9.505 

1.425 

Sg 

3.168 

8.132  ' 

1%  10.983 

1% 

0.455 

X 

1.901 

% 

3.802 

ik 

8.871 

1%  111.828 

1.406 

WEIGHTS  OF    FLAT    BAR    IRON. 


283 


TABLE—  Continued. 


.2 

c 

£ 

c 

a 

& 

.2     2 

| 

.S 

c 

& 

c 

.S 

£ 

.2 

c 

a 

.2^ 

.2 

•s  i  i 

.2 

d 

c 

a 

e 

c 

C 

i 

g 

JL 

g 

i 

g 

fl 

g 

ji 

H 

g_ 

i 

I 

g 

N 

12.356 

P' 

12.199 

2M 

26.719 

m 

19.221 

iX. 

17.953 

''i 

13.307 

LK 

13.308 

K 

1.267 

20.699 

21.544 

/8 

14.257 

14.417 

X 

2.535 

1% 

22.178 

L% 

25.135 

15.208 

1  '"•:{ 

15.526 

3.802 

2 

23.656 

! 

28.725 

^K 

16.158 

\\- 

16.635 

jy 

5.069 

2X 

26.613 

32.316 

1% 

X 

1.003 

2 

17.744 

k< 

6.337 

2  '  . 

29.570 

21/ 

35.907 

2.006 

21- 

18.853 

X 

7.604 

ix 

32.527 

iy 

39.497 

/8 

3.009 

JU 

19.962 

% 

8.871 

3 

35.485 

3 

43.088 

K 

4.013 

2:i.. 

21.071 

1 

10.138 

3X 

38.441 

3X 

46.679 

;S8 

5.016 

2'1'.', 

22.180 

Ilx3 

11.406 

3% 

K 

1.584 

50.269 

6.019 

2% 

K 

1.162 

IX 

12.673 

X 

3.168 

3V 

63.860 

% 

7.022 

X 

2.323 

13940 

% 

4.752 

4 

57.450 

8.025 

% 

3.485 

IK 

15.208 

K 

6.336 

<%. 

X 

3.802 

K 

9.028 

K 

4.647 

1  5  .. 

16.475 

% 

7.921 

7.604 

10.032 

/« 

5.808 

1?4 

17.742 

M 

9.505 

/•I 

11.406 

L% 

11.035 

% 

6.970 

1% 

19.010 

/6 

11.089 

1 

15.208 

'•K 

12.038 

% 

8.132 

2 

20.27r 

1 

12.673 

IX 

19.010 

[•;£ 

13.042 

1 

9.294 

2X 

22.811 

IK 

14.257 

22.812 

[% 

14.046 

IK 

10.455 

2  \;z 

25.346 

IX 

15.841 

i  ';  r 

26.614 

1% 

15.048 

IX 

11.617 

2/4 

27.881 

1% 

17.425 

2 

30.415 

2 

16.051 

1.3.;, 

12.779 

3X 

i,8 

1.373 

IK 

19.009 

2K 

34.217 

2/8 

17.054 

IK 

13.940 

X 

2.746 

^/8 

20.594 

38.019 

w  '4 

18.057 

1-TjT 

15.102 

/3<! 

4.119 

1  ;*,4 

22.178 

L'  '•'•  ^ 

41.820 

2K 

K 

1.056 

i  ;V 

16.264 

4jj 

5.492 

]% 

23.762 

3 

45.623 

X 

2.112 

1% 

17.425 

?i 

6.865 

2 

25.346 

8X 

49.425 

/« 

3.168 

2 

18.587 

/-'t 

8.237 

ax 

28.514 

53.226 

^ 

4.224 

2X 

19.749 

/« 

9.611 

31.682 

'"'  ;i'-4 

57.028 

K 

5.280 

2  '4 

20.910 

1 

10.983 

2X 

34.85r 

4 

60.830 

X 

6.336 

2% 

22.0724 

1^8 

12.356 

ar 

38.019 

4X 

64.632 

7.392 

23.234 

l'1  1 

13.730 

3^ 

41.187 

4M 

X 

4.013 

l/8 

8.448 

v..,, 

24.395 

1  '-',.. 

15.102 

44.355 

8.026 

IK 

9.504 

2% 

K 

1.215 

I  IK 

16.47 

1 

2 

1.690 

X 

12.039 

IX 

10.560 

X 

2.429 

1  'f] 

17.84c 

X 

3.380 

1 

16.052 

1/^3 

11.616 

% 

3.64- 

]  :'. 

19.22 

6.759 

IX 

20.066 

IK 

12.672 

4.85^- 

IJ'H 

20.59 

X 

10.138 

IX 

24.079 

13.728 

% 

6.07:. 

2 

21.96 

1 

13.518 

1  •';.  j 

28.093 

i  •••! 

14.781 

sv 

7.287 

2X 

24.71 

^ 

16.897 

2  ' 

32.105 

iji 

15.840 

% 

8.502 

"-'/•••: 

27.45 

20.27 

-X 

36.118 

2 

16.896 

1 

9.7K 

2% 

30.20 

1  l\i 

23.65 

2  A  , 

40.131 

2.'; 

17.952 

IX 

10.931 

3 

32.950 

2 

27.03 

2X 

44.144 

^''1 

19.00h 

iV,' 

12.145 

'•'>  l. 

K 

1.471 

2X 

30.41 

3 

48.157 

2# 

^ 

20.06 
l.lOf 

13.360 
14.574 

X 

2.95 
4.436 

2>^  !  33.79 
2%  1  37.17 

3¥ 

52.170 
56.184 

X 

2.218 

ifl 

15.789 

/" 

5.91 

3 

40.55 

•';  "'» 

60.197 

/i 

3.327 

i  •?( 

17.003 

/i" 

7.39 

HX 

43.93 

4 

64.210 

* 

4.4,% 

i7,. 

18.218 

x-i 

8.87 

:•;>, 

47.31 

4):i 

68.223 

5.545 

2 

19.43i, 

r/ 

10.350 

50.69 

•11. 

72.235 

X 

6.654 

20.647 

1 

11.82 

*x 

IB 

1.79 

5 

X 

4.224 

% 

7.763 

Likj 

21.861 

1  i,a 

13.30 

X 

3.59 

8.449 

i 

8.872 

-);;i; 

23.076 

1  ^ 

14.78, 

k 

7.18 

/i 

12.673 

IK 

9.981 

Jl., 

2-1.200 

1  -i  j, 

16.26 

10.772 

1 

16.897 

IX 

11.090 

88 

25.50, 

1^-. 

17.74 

l"1 

14.364 

IX 

21.122 

284 


RA1NEY  S    IMPROVED    ABACUS. 


Weight  of  a  lineal  foot  of  Cast  Iron  Pipes,  or  Cylinders,  in  Ibs., 
the  thickness  and  bore  being  given. 


£c 

S.S 

£.2 

8.S 

s  c 

•2^ 

s^ 

2.S 

d  ^ 

£J2 

S.S 

J.s 

£5 

D  '" 

tt.S 

3    g 

11 

£* 

11 

£j 

?LS 

11 

IS 

«! 

£r 

il 

1 

X 

3.06 

* 

43.68 

% 

77.36 

17  X 

x 

88.23 

% 

5.05 

% 

53.30 

% 

93.70 

% 

111.06 

IX 

X 

3.67 

63.18 

% 

110.48 

% 

134.16 

% 

6. 

7 

& 

36.66 

1 

127.42 

% 

157.59 

IX 

K 

6.89 

% 

46.80 

12  X 

X 

63.70 

i 

181.33 

X 

9.80 

% 

56.96 

% 

80.40 

18 

% 

114.10 

1% 

% 

7.80 

% 

67.60 

% 

97.40 

X 

137.84 

% 

11.04 

i 

78.39 

% 

114.72 

% 

161.90 

2 

% 

8.74 

7  n., 

% 

39.22 

1 

132.35 

i 

186.24 

X 

12.23 

% 

49.92 

13 

X 

66.14 

19 

% 

120.24 

2X 

% 

9.65 

K 

60.48 

% 

83.46 

% 

145.20 

X 

13.48 

% 

71.76 

% 

101.08 

% 

170.47 

2X 

% 

10.57 

1 

83.28 

% 

118.97 

i 

195.92 

% 

14.66 

8 

X 

41.64 

1 

137.28 

20 

% 

126.33 

% 

19.05 

% 

52.68 

13  X 

X 

68.64 

% 

152.53 

2% 

% 

11.54 

% 

64.27 

% 

86.55 

% 

179.02 

X 

15.91 

% 

76.12 

X 

104.76 

i 

205.80 

% 

20.59 

1 

88.20 

% 

123.30 

21 

K 

132.50 

3 

% 

12.28 

8X 

X 

44.11 

1 

142.16* 

X 

159.84 

K 

17.15 

% 

56.16 

14 

X 

71.07 

% 

187.60 

% 

22.15 

K 

68. 

% 

89.61 

i 

215.52 

X 

27.56 

% 

80.50 

% 

108.46 

22 

% 

138.60 

3X 

i/ 

18.40 

1 

93.28 

/'8 

127.60 

X 

167.24 

fc 

23.72 

9 

X 

46.50 

1 

147.03 

» 

196.46 

X 

29.64 

% 

58.92 

'4>a 

X 

73.72 

i 

225.38 

3X 

X 

19.66 

% 

71.71 

% 

92.66 

i3 

% 

144.77 

X58 

25.27 

% 

84.70 

X 

112.10 

% 

174.62 

X 

31.20 

1 

97.98 

% 

131.86 

K 

204.78 

3M 

X 

20.90 

9X 

X 

48.98 

1 

151.92 

1 

235.28 

x5** 

26.83 

P 

62.02 

15 

X 

75.96 

24 

% 

150.85 

% 

33.07 

& 

75.32 

% 

95.72 

X 

181.92 

4 

x 

22.05 

% 

88.98 

X 

115.78 

% 

213.28 

% 

28.28 

1 

102.90 

% 

136.15 

i 

245.08 

X 

34.94 

10 

X 

51.46 

1 

156.82 

% 

156.97 

4X 

X 

23.35 

% 

65.08 

15  K 

1^ 

78.40 

K 

180.28 

/'8 

29.85 

% 

78.99 

t/ 

98.78 

% 

221.94 

X 

36.73 

% 

93.24 

119.48 

i 

25-1.86 

4X 

X 

24.49 

1 

108.84 

J? 

140.40 

16 

% 

196.62 

/5/8 

31.40 

10  % 

X 

53.88 

1 

161.82 

% 

230.56 

% 

38.58 

% 

68.14 

16 

X 

80.87 

i 

264.66 

4% 

X 

25.70 

X 

82.68 

% 

101.82 

27 

% 

204.04 

% 

32.91 

% 

97.44 

K 

123.14 

% 

239.08 

X 

40.43 

1 

112.68 

% 

144.76 

1 

274.56 

5 

X 

26.94 

11 

X 

56.34 

1 

166.60 

28 

M 

211.32 

% 

34.34 

71.19 

ie  yz 

X 

83.20 

% 

247.62 

% 

42.28 

M 

86.40 

% 

104.82 

1 

284.28 

SK 

X 

29.40 

% 

101.83 

X 

126.79 

29 

X 

218.70 

% 

37.44 

1 

117.60 

% 

149.02 

% 

256.20 

% 

45.94 

ix 

X 

58.82 

1 

171.60 

i 

294.02 

X 

31.82 

% 

74.28 

17 

K 

85.73 

80 

% 

226.20 

% 

40.56 

K      90.06 

% 

107.96 

% 

264.79 

X 

49.60 

%    106.14 

X 

130.48 

i 

303.86 

% 

58.96 

1        122.62 

% 

153.30 

X 

343.20 

^ 

X 

34.32 

12 

V*      61.26 

i 

176.58 

GOLD  COIN. —  United  States,  English  and  French.    285 


4  Eagle  ri7951._$5.25.  Eagle.- $10.50.  %  Eagle.— $5,00. 


Eagle— $2,50.  Sovereign.-— $4,85 


286        GOLD  COIN. — American,  English  and  French. 


40  Francs.— $7,60.  Double  Sovereign.— $9,60.     Eagle.— $10,00. 


Bechtler  pieces.— $2.37.  20  Francs.— $3  83     Sover'n. — $2,42- 


American  Gold  Dollar.  16th  Doubloons.-  .$1.  #1.20. 


Sovereign. — $4,85.      Double  Louis  d'or. — $8,50.      Sovereign.-^$4£5. 


GOLD  Go™.— English,  French  and  South  American.    287 


Sovcreign.-$4,85.          t  DoUbloou.-S3.75. 


4DoabIooB.-f7.7S. 


288         GOLD  COIN. — French  and  South  American. 


Doubloo  i.—  $15,55-60.  Doubloon.— $15,50. 


20  Francs.— $3.83.         4  Douoioon.-  $4.  Sovereign.— $4.85 


GOLD  COIN. — South  American  and  German.  289 


^840 

Doub.— $3,-0.        £  Doubloon. -$7.75.         Double  Frecl.d'or.— $7.80. 


Moidore  [Brazil]  $4,80.  $  Joe.— $8.50. 


$1,96 


290 GOLD  COIN, — S.  American,  Portuguese,  Spanish,  German. 


Doubloon. — $15,75-16. 


Doubloon.— $15,55-60. 


Escudo.—  $1,90.       Moidore.  —  iOc.  .     24  cents. 


£  Thai  —$1,94. 


Doubloon. — $15.55. 


10  Thaler.- -$780 


10  Thaler.-$7.80. 


10Thaler.-|7.80. 


40  Lire.-$7.66 


GOLD  COIN. — Portuguese,  German  and  Italian.       291 


One  Mohur.— $6.7         10  Tlmlet.-f  7.80 


1-16  Dou.— lOcts.          10th  Moidore.— 50  cents.          1-16  Doub.— 90cts. 


10  Thalers— $7,80.         1  )  Timie.s— $7  80.         10  Thalers— $7,80. 


Fred.d'or.— 3.90. 


5  Thale7.~ |3.90. 


-"l*^ 
5  Tlialers— $3,10 


292 


GOLD  COIN. — German  and  Italian. 


Double  Ducat —$4.40  lOScudo— $10  Soverain.— $6.50. 

^ 


20  Lire  -$3  80      10  Lire.— €  1.90      5  Gilder. — $2  Ducat.— $2.20. 


80  Lire.— $  1 5.32.  5  Thaler— $3.90. 


20  Lire.— $:3.80.         \  Joe.— §1.75.       Fvquni.— $2.20.        Ducat  .^2.20, 


Ducat.— 12.20 


20  Lire.— 83.80 


DUCUL  —$2.20 


GOLD  COIN. —  German  and  ltd.     SILVER  COIN. —  U.  #293 


Gold  Crown— -$5,72.          >£  Imperial— $3,90.         5  Roubles.— $3,90. 


10  Guilders— $3,95.  20  Lire.— $3,82.  2  Christian  u'or  $7,80. 

U*  States   Silver    Coins. 


I  dime,  5  cents. 


U.  States— One  Dollar. 


One  Dime,  10  cte. 


SILVER  COIN. —  United  States 


Quarter  Dollar,  25  cents,        %  Dollar.     25  cents.          *  Dollar,  25  cents 


SILVER  COIN. —  United  States  and  English.  295 


Dollar.    25  cents.  Dime.    10  cents.        7  cent 


Shilling— 20  cents.  ^  Dollar— 24  cents.  Shilling,  20  cents. 


SILVER  COIN. — English. 


25  cents.  1  Shilling     18  cents.       Shilling.     20  cents. 


I  Shilling.    19  cents.  18  cents.  1  Shilling.    18  cents 


SILVER  COIN. — English. 


1  Rupeo.     40  cents.  U  Dollar.  lOcts. 


I  H  ISH 

**SgnJ!^ 
11  cents  Six-pence,  10  cents.  5  Cent3. 


SILVER  COIN.—  English  and  Spanish. 


_  _ 

Dollar.    10  cts        }£  Pistareen.  8  cts-        Real.    10  cents. 


VB   17420 


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